elliptic special weingarten surfaces of minimal type with ... · este trabajo no hubiese sido...

Elliptic Special Weingarten Surfaces of Minimal

Type with Finite Total Second Fundamental Form

Héber Mesa Palomino

April, 2019

Instituto Nacional de Matemática Pura e Aplicada

Doctoral Thesis

Elliptic Special Weingarten Surfaces of Minimal

Type with Finite Total Second Fundamental Form


Thesis presented to the Post-graduate Program in Mathematics at

Instituto Nacional de Matemática Pura e Aplicada as partial fulfillment

of the requirements for the degree of Doctor in Mathematics.

1. Reviewer Marco GuaracoUniversity of Chicago

2. Reviewer Asun JiménezUniversidade Federal Fluminense

3. Reviewer Harold RosenbergInstituto Nacional de Matemática Pura e Aplicada

4. Reviewer Luis FloritInstituto Nacional de Matemática Pura e Aplicada

5. Reviewer Walcy SantosUniversidade Federal do Rio de Janeiro

Advisor José María Espinar GarcíaInstituto Nacional de Matemática Pura e Aplicada

Universidad de Cádiz

April, 2019


Elliptic Special Weingarten Surfaces of Minimal Type with Finite Total Second Fundamental Form

Doctoral Thesis, April, 2019

Reviewers: Marco Guaraco, Asun Jiménez, Harold Rosenberg, Luis Florit and Walcy Santos

Advisor: José María Espinar García

Instituto Nacional de Matemática Pura e Aplicada

Estrada Dona Castorina, 110

CEP 22460-320

Rio de Janeiro, RJ-Brazil

To Yulieth, my wife... my whole family

Agradecimientos

La razón por la que ésta parte del documento se encuentra escrita en español esporque espero lograr transmitir a todas las personas que tuvieron alguna contribu-ción en mi proceso de formación, la gratitud que siento por los momentos vividos enestos últimos años que me han hecho crecer como persona y como matemático.

Primero que todo agradezco a Dios por brindarme los medios necesarios paracumplir las metas que me he propuesto a lo largo de la vida. Definitivamente,la culminación de este proyecto es para mi una bendición que se suma a muchasotras.

A Yulieth, agradezco su amor incondicional y su valentía para acompañarme enesta aventura, en la que aprendimos que nuestro hogar no estaba en ningún lugarmás que en nosotros mismos, que no importaba nada siempre que estuviésemosjuntos. Gracias por soportar tantos cambios en estos últimos seis años y por haceruna pausa en tus sueños para ayudarme a cumplir los míos.

Este trabajo no hubiese sido posible sin la directriz del profesor José Espinar, miorientador. Sus observaciones fueron tan puntuales y precisas que no me quedamás que admirar y respetar su profesionalismo. Las diferentes charlas que tuvimosmientras fumaba un cigarro y tomábamos café, las recuerdo con mucho apreciopues gran parte de lo logrado en este trabajo se debe a aquellas tertulias. Quedo endeuda con su paciencia, ya que soy consciente de que varias de sus orientacionesme tomaron tiempo para asimilarlas.

Agradezco a los profesores Marcelo Viana y José Espinar por aceptarme en suproyecto de escribir un libro de Ecuaciones Diferenciales. Ha sido muy grato contarcon su confianza para diseñar a mi antojo las gráficas del libro, estoy muy felizde la matemática, la programación y los macros de PSTricks que aprendí en esteproceso.

Mi experiencia en Rio fue, en exceso, agradable. La razón, haber tenido la suertede conocer personas que enriquecieron mis experiencias. Los profesores Luis Flority Henrique Bursztyn me ofrecieron los mejores cursos que recibí en el IMPA. Miscompañeros de geometría, Haimer Trejos y Ángela Roldán, me transmitieron suadmiración por mi orientador. Inocencio Ortíz, compartió conmigo todo el sufri-miento de los exámenes de calificación, parte de mi éxito en esa etapa se la deboa su amistad y dedicación de estudio (toda una maratón geométrico-topológica).

vii

Guilherme Goedert, uno de mis pocos amigos brasileños, me acercó al mundo de laprogramación y fue el compañero perfecto en el trabajo del libro de Ecuaciones Dife-renciales. Inocencio Ortíz, Laura Molina, Daniel Rodriguez, Betina Moura, DavidAndrade, Midory Quispe, Plinio G. P. Murillo, Vanessa Rodrigues, Jhovanny Muñozy Ana Medina, compartieron con nosotros años llenos de alegrías, juegos, comiday bebidas, nunca hubo una mala excusa para reunirnos, dejar pasar el tiempo yver el amanecer. Rafael Sanabria y Adriana Sánchez, ‘los patrones’, ‘los ticos’, noshicieron la vida más fácil con su amistad, son los responsables directos del exceso.Don Edson Reis, un amigo carioca, que con la excusa de aprender algo de españolnos brindó su amistad y nos hizo parte de su día a día en Parque da Gávea.

Nuestra estancia en Brasil también se vio favorecida por el apoyo de muchaspersonas en Colombia. Agradezco a Katherine Moreno por seguirnos brindandosu amistad a pesar de la distancia, en especial para con Yulieth, y siempre estarpendiente de nosotros, abasteciéndonos de suficiente café colombiano. Igualmenteagradezco a todos los docentes del Departamento de Matemáticas que finalmenteasumieron mi carga docente durante los años de mi comisión académica; en espa-cial a la profesora Ana María Sanabria que ha confiado ciegamente en mis capaci-dades para culminar este proyecto.

Finalmente debo agradecer al IMPA y todos sus funcionarios por su calidez. AlConselho de Desenvolvimento Científico e Tecnológico (CNPq) y la Universidad delValle por el financiamiento económico, sin el cuál no hubiese sido posible dedicarmi tiempo al estudio y la investigación.

Héber Mesa PalominoApril, 2019

Abstract

The aim of this work is to study Elliptic Special Weingarten Surfaces of MinimalType in the Euclidean space R3, ESWMT-surfaces in short, that is, those whosemean curvature, H, and Gaussian curvature, K, satisfy a relation of the formH = f(H2 − K), where f : [0,∞) → R is a continuous function of class C1 in(0,+∞) satisfying

4t(f ′(t))2 < 1 for all t ∈ (0,∞) and f(0) = 0.

Our main result establishes sufficient conditions for an embedded ESWMT-surfaceto be rotationally symmetric. This result, together with the characterization of ro-tationally symmetric examples due to Sa Earp and Toubiana, yields a Schoen typetheorem for ESWMT-surfaces. Specifically, we prove

Main Theorem. LetM be a complete connected ESWMT-surface embedded

in R3 with finite total second fundamental form and two ends. Suppose

that the function f associated to M is non-negative and Lipschitz. Then

M must be rotationally symmetric. In fact, there exists τ > 0 such that

M is the surface of revolution Mτ determined by Sa Earp and Toubiana in

[ST95].

In order to obtain this result, we analyze the asymptotic growth of the heightfunction on the ends of an ESWMT-surface using a conformal parameter and then,using the Kenmotsu representation formula, we are able to describe these ends asgraphs of functions of logarithmic order. Finally, we use the Aleksandrov reflectionmethod through tilted planes to obtain the rotational symmetry.

As a corollary of the main theorem, we can recover the original Schoen theorem,namely,

Corollary. Let M be a complete connected embedded minimal surface in R3 with finite

total curvature and two ends. Then M is a catenoid.

ix

Resumo

O objetivo deste trabalho é estudar as Superfícies Elípticas Especiais de Weingartendo Tipo Mínimo no espaço euclideano R3, também chamadas ESWMT-superfíciesem referência à nomenclatura inglesa, isto é, aquelas que sua curvatura meia, H, ea curvatura Gaussiana, K, satisfazem uma relação da forma H = f(H2 −K), ondef : [0,∞) → R é uma função contínua de classe C1 em (0,+∞) satisfazendo

4t(f ′(t))2 < 1 para todo t ∈ (0,∞) e f(0) = 0.

Nosso principal resultado estabelece condições suficientes para que uma ESWMT-superfície mergulhada seja rotacionalmente simétrica. Este resultado, junto coma caracterização dos exemplos rotacionalmente simétricos devidos a Sa Earp eToubiana, resulta em um teorema do tipo Schoen para ESWMT-superfícies. Especi-ficamente provamos que,

Teorema Principal. Seja M uma ESWMT-superfície conexa, completa e

mergulhada em R3 com segunda forma fundamental total finita e dois

fins. Suponha a função f associada à M é não negativa e lipschitziana.

Então M deve ser rotacionalmente simétrica. De fato, existe τ > 0 tais que

M é a superfície de revolução Mτ determinada por Sa Earp e Toubiana em

[ST95].

Para obter este resultado, nós analisamos o crescimento assintótico da funçãoaltura nos fins duma ESWMT-superfície usando um parâmetro conforme e então,usando a formula de representação de Kenmotsu, fomos capazes de descrever osfins como gráficos de funções de ordem logarítmica. Finalmente usamos o métodode reflexão de Aleksandrov através de planos inclinados para obter a simetria rota-cional.

Como corolário do teorema principal, podemos recuperar o teorema original deSchoen, a saber,

Corolário. Seja M uma superfície mínima conexa completa e mergulhada em R3 com

curvatura total finita e dois fins. Então M é um catenoide.

xi

Contents

Abstract ix

Resumo xi

Introduction xv

1 Preliminaries 1

1.1 Fundamental forms of an immersion . . . . . . . . . . . . . . . . . . 21.2 The Gauss map and the representation of surfaces in R3 . . . . . . . 61.3 Finite total curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Maximum principles and geometric applications . . . . . . . . . . . . 121.5 Codazzi pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Elliptic Special Weingarten Surfaces of Minimal Type 21

2.1 General facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Rotational examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Special Codazzi pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 The Gauss map of Special Weingarten surfaces . . . . . . . . . . . . . 27

3 Asymptotic behaviour of ends of ESWMT-surfaces 33

3.1 Regular at infinity for minimal surfaces . . . . . . . . . . . . . . . . . 333.2 ESWMT-surfaces with finite total second fundamental form . . . . . . 343.3 Growth of an ESWMT-end . . . . . . . . . . . . . . . . . . . . . . . . 363.4 The height near infinity . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Characterization of ESWMT-surfaces with two ends 55

4.1 Embedded ESWMT-surfaces with two ends . . . . . . . . . . . . . . . 554.2 The Aleksandrov function and tilted planes . . . . . . . . . . . . . . . 564.3 A Schoen type theorem for ESWMT-surfaces . . . . . . . . . . . . . . 60

Bibliography 63

xiii

Introduction

Two of the most remarkable theorems in the study of differential geometry in thelarge of surfaces immersed in the Euclidean space R3 are the Liebmann theorem,asserting that an immersed sphere with constant Gaussian curvature is a roundsphere; and the Hopf theorem, which states that an immersed sphere with cons-tant mean curvature is a round sphere. Constant Gaussian curvature surfaces andconstant mean curvature surfaces can be seen as particular cases of a bigger familyof surfaces, the so-called Weingarten surfaces.

Generally speaking, a Weingarten surface (W-surface) is a surface immersed ina three-dimensional manifold such that the principal curvatures, k1 and k2, of theimmersion satisfy certain relation, i.e., there is a smooth function W of two varia-bles such that W (k1, k2) = 0, see [Che45; HW54]. This family of surfaces wasintroduced by Weingarten in the context of finding all surfaces isometric to a givensurface of revolution, see [Wei61; Wei63]. Note that a Weingarten relation can berewritten (generically) as a relation between the mean curvature, H, and the Gau-ssian curvature, K, of the surface, which we denote in the same way, W (H,K) = 0.Closed Weingarten surfaces in R3 were widely studied by Hartman and Wintner[HW54], Chern [Che45; Che55b], Hopf [Hop83] and Bryant [Bry11] obtaining,among others, a generalization of Hopf’s theorem and Liebmann’s theorem.

When the function W is linear, that is, when we are considering a Weingartenrelation of the form aH + bK = c, where a, b, c are real constants, W-surfaces arecalled Linear Weingarten surfaces, abbreviated by LW-surfaces. These surfaces havebeen studied by several authors. For instance, Rosenberg and Sa Earp in [RS94]showed that if b = 1 and a, c > 0, the annular ends of a properly embedded LW-surface converge to Delaunay ends. Gálvez et al. in [Gá+03] obtained a harmonicrepresentation of LW-surfaces of elliptic type, and optimal height and curvature es-timates, moreover, they characterized the spherical caps as the only LW-surfaces ofelliptic type achieving these bounds. On the other hand, López gave a classificationof rotational LW-surfaces of hyperbolic type, see [Ló08b]. Immersed LW-surfacesin other ambient spaces have been also analyzed, to mention, Aledo and Espinarstudied LW-surfaces in the de Sitter space reaching a classification of complete LW-surfaces with non-negative Gaussian curvature, see [AE07]; and Barros et al. in[Bar+12] presented a complete description of all rotational LW-surfaces in the Eu-clidean sphere S3.

xv

It is a special case when the Weingarten relation W (H,K) = 0 can be writtenas H = f(H2 − K), where f : [0,∞) → R is a continuous function of class C1

in (0,+∞). In this case, the W-surfaces are called Special Weingarten surfacesor simply SW-surfaces and we will refer to f as the associated function of theSW-surface. An interesting class of SW-surfaces is obtained when the function f

satisfies 4t(f ′(t))2 < 1 for all t ∈ (0,∞). This class of surfaces is called SpecialWeingarten surfaces of elliptic type and denoted by ESW-surfaces or f -surfaces, see[ST93; ST99b; ST01; Ale+10]. Rosenberg and Sa Earp in [RS94] proved that thereexists a large family of LW-surfaces contained in the family of ESW-surfaces, namelywhen b = 1, a ≥ 0 and c > 0. One of the most useful results about the family ofESW-surfaces in R3 is that it satisfies the maximum principle, a result achieved byRosenberg and Sa Earp in [RS94], and Braga and Sa Earp in [BS97]. Aledo etal. in [Ale+10] classified ESW-surfaces for which the Gaussian curvature does notchange sign, obtaining that, if an ESW-surface is complete with K ≥ 0 then it mustbe either a totally umbilical sphere, a plane or a right circular cylinder, and, if anESW-surface is properly embedded with K ≤ 0, then it is a right circular cylinderor f(0) = 0. In [Gá+16], Gálvez et al. improved the above result in the case K ≤ 0

removing the properly embedded hypothesis and assuming completeness.

The family of ESW-surfaces can be divided in two classes, namely f(0) 6= 0 andf(0) = 0. In the first case, ESW-surfaces are called of constant mean curvaturetype, and in the second case, are called Elliptic Special Weingarten surfaces of mi-nimal type, ESWMT-surfaces in short. For the first family of surfaces, Aledo et al. in[Ale+10] extended the results obtained in the theory of constant mean curvaturesurfaces by Korevaar et al. in [Kor+89], specifically, they showed that if an ESW-surface of constant mean curvature type is properly embedded, in R3 or H3, hasfinite topology and k ends, then, k ≥ 2, is rotationally symmetric if k = 2, and iscontained in a slab if k = 3. On the other hand, about ESWMT-surfaces, Sa Earpand Toubiana in [ST95] proved existence and uniqueness of rotational symmetricESWMT-surfaces. They also proved, under some additional condition on the func-tion f , that there exists an one-parameter family of ESWMT-surfaces that behavesas the family of minimal catenoids, allowing them to prove a half-space theoremin this context. ESWMT-surfaces in other ambient spaces have been studied byMorabito and Rodríguez in [MR13], where they established necessary and sufficientconditions for existence and uniqueness of complete rotational ESWMT-surfaces inS2 × R and H2 × R and showed the existence of non-complete examples of suchsurfaces. Also, de Almeida et al. proved in [de +09] the existence of rotationalESWMT-surfaces in S3.

Trivially, any plane is an ESWMT-surface. This basic fact together with the maxi-mum principle make us think in the theory of ESWMT-surfaces in the same way asthe theory of minimal surfaces. Hence, developing the understanding of ESWMT-surfaces we are contributing to improve the knowledge of a broader class of sur-

xvi

faces. With this premise in mind, it is natural to ask: which of the classical theoremsof minimal surfaces in R3 can be extended to ESWMT-surfaces?

The theory of minimal surfaces is huge, nevertheless we will focus for a momenton complete minimal surfaces with finite total curvature in R3. Finite total curva-ture means that the Gaussian curvature, K, is integrable on M , i.e.,

∫

MK dA < ∞.

A classical result, due to Chern and Osserman in [CO67], states that each of thesesurfaces is of finite topological type. Specifically, it is conformally equivalent to acompact Riemann surface with a finite number of points removed. These pointsare the representation of the ends of the minimal surface. In particular, each end isconformally a punctured disk. Moreover, the Gauss map of the immersion extendsto a meromorphic function defined over the whole disk, which means that the Gaussmap can be defined globally over the compact Riemann surface associated to theminimal surface. Jorge and Meeks III, in [JM83], proved that a surface of finitetopological type, such that the Gauss map is well defined at infinity, is properlyembedded and, viewed from infinity, looks like a finite collection of planes passingthrough the origin. A nice way to interpret this result is to think that the endsof such surfaces behave like the ends of either a catenoid or a plane at infinity.Using this result they showed that an embedded minimal surface with finite totalcurvature must be a flat plane or it has at least two ends; in particular, if the surfaceis not a plane, it cannot be simply connected. Furthermore, they proved that thetotal curvature of a complete minimal surface M with finite total curvature, Eulercharacteristic χ(M) and k ends must satisfies

∫

MK dA = 2π (χ(M) − k)

if, and only if, the ends are embedded. As a consequence of this formula, they ob-tained a characterization of the catenoid as the unique embedded minimal surfacewith finite total curvature, two ends and genus zero.

The characterization obtained for the catenoid by Jorge and Meeks III was im-proved by Schoen in [Sch83] assuming the embedding hypothesis just for the endsand without any assumption on the topology of the surface. More precisely, heproved that a complete minimal immersion of a hypersurface in Rn, which is regu-lar at infinity and has two ends, must be a pair of hyperplanes or a catenoid. Theconcept of regular at infinity defined by Schoen means, roughly speaking, that eachend of the surface is well behaved in the sense that the “growth at infinity” fromsome fixed plane is controlled and almost radial with respect to some axis. In otherwords, that the ends are quite similar to the ends of either a catenoid or a plane.The precise definition will be presented in Chapter 3. In the Euclidean space R3,Schoen proved that the property of being regular at infinity for an immersed mini-

xvii

mal surface M is equivalent to finite total curvature and embedded ends. There isa version of Schoen’s theorem in H2 × R due to Hauswirth et al., see [Hau+15].In this paper, they proved that a complete minimal surface immersed in H2 × R

with finite total curvature and two ends, each one asymptotic to a vertical geodesicplane, must be a horizontal catenoid.

Taking into account the existence and uniqueness of revolution ESWMT-surfaces,we pursuit to obtain a Schoen type theorem, which establishes these examples asthe only ones with finite total curvature.

Schoen’s proof can be described in three parts.

• As we have mentioned, complete minimal surfaces with finite total curvatureare of finite topological type and the Gauss map extends conformally to infi-nity, see [CO67; Oss69; JM83].

• There is a well-known link between minimal surfaces and complex analysis.This relation is given by the Weierstrass representation, which says that anyminimal surface can be represented by complex holomorphic functions, calledWeierstrass data. By Osserman’s results, the Weierstrass data can be extendedto the ends as meromorphic functions. Schoen used these extensions and theWeierstrass representation to derive an asymptotic expansion for embeddedends, concluding that each end must be geometrically asymptotic either to aplane or to a catenoid. By applying the flux formula he proved that if the sur-face is connected and has two ends, then the ends should be parallel, and bythe half-space theorem, both ends have (simultaneously) logarithmic growth,in opposite directions.

• Finally, Schoen used the Aleksandrov reflection method to prove that the sur-face has a horizontal plane of symmetry and a vertical axis of symmetry. Thismeans that the minimal surface is rotationally symmetric, hence a catenoid ora pair of planes.

As we can see, Schoen’s proof relies strongly on the fact that the surface is mini-mal and on the connection of the theory of minimal surfaces with complex analysis.This reasoning cannot be translated directly to ESWMT-surfaces because these sur-faces have no natural connection with complex analysis. This is one of the obstaclesto overcome. We are going to introduce some techniques that helped us to use asimilar scheme as Schoen, but avoiding the minimal hypothesis.

The first point to deal with is the finite total curvature hypothesis. Thanks tothe work of Huber it is well known that surfaces with finite total curvature has anatural compactification, see [Hub57]. In this paper, he proved that a completesurface with finite total curvature is isometric to a compact Riemannian surfacewith finitely many singular points, each admitting a neighborhood isometric to thepunctured disk equipped with a conformal flat metric, see [HT92]. Unfortunately,the hypothesis of finite total curvature does not imply that the Gauss map can be

xviii

extended to the singularity, that is, the Gauss map can have an undesirable behaviorat infinity, see [Whi87; Con+15] for examples of such kind of ends. To settle thissituation, we consider a stronger assumption namely finite total second fundamen-tal form, which means that the norm of the second fundamental form, |II|, of animmersed surface M in R3, is L2 on the surface, that is,

∫

M|II|2 dA < ∞.

White proved in [Whi87] that if a surface has finite total second fundamental form,then it has finite total curvature and also the Gauss map extends continuously toinfinity. Furthermore, it can be proved that if the end is embedded, then the end isa graph over a plane orthogonal to the Gauss map at infinity. White asked whetherfinite total second fundamental form implies properly immersed, this was answeredpositively by Müller and Ševerák in [MŠ95]. Hence, for ESWMT-surfaces, we willadopt finite total second fundamental form as a natural hypothesis.

Once we have adopted the finite total second fundamental form hypothesis, thesecond point to deal with is the growth of an embedded end. In 1915, Bern-stein proved that a C2 function u(x, y) defined in the whole x, y-plane such thatuxxuyy − u2

xy ≤ 0 and uxxuyy − u2xy 6≡ 0 cannot be bounded, see [Ber15]. The

original proof had a gap of topological nature. Hopf, in 1950, bridged this gap andimproved the result of Bernstein, see [Hop50a; Hop50b]. The theorem of Hopfestablishes that a function with the conditions imposed by Bernstein cannot be o(r)where r is the distance from an arbitrary point to a chosen fixed point. The proofpresented by Hopf was adapted by Chan to prove that there is no isometric immer-sion in R3 from a complete oriented, one ended, non-positively curved surface withan isolated set of parabolic points and finite total second fundamental form, suchthat the end is embedded, see [Cha00]. We adapt Chan’s proof to demonstratethat if the graph of a real valued function, defined outside a compact subset of R2,is an ESWMT-surface with finite total second fundamental form, then the functiongrows infinitely just in one direction, that is, it has either a finite lower or an upperbound. This fact may seem natural by the hypothesis on the curvatures, however,we can find in the literature examples of surfaces with finite total curvature thathave highly complicated ends, as for instance the examples exhibited by Connoret al. in [Con+15].

The study of the Gauss map of an immersed surface in R3 by isothermal coordi-nates can derive in representation formulas analogous to the Weierstrass represen-tation, see [Ken79; GM00]. This was the idea developed in [Ken79] by Kenmotsuto obtain a representation formula for immersed surfaces of prescribed mean cur-vature by means of the Gauss map. Nevertheless, since the Gauss map just extendscontinuously to singularities, the functions involved in the Kenmotsu representationmay not have an extension to infinity as meromorphic functions.

xix

Using the fact that ends of ESWMT-surfaces with finite total second fundamentalform cannot grow infinitely in opposite directions, we are able to use some resultsof Taliaferro about the growth of a positive superharmonic function near an isolatedsingularity, see [Tal99; Tal01a], for obtaining a Bôcher type theorem that gives usa full description of the height function defined on a conformal parameter. In or-der to use the Kenmotsu representation, it is necessary to have some additionalinformation about the Gauss map. It is known that the Gauss map, g, satisfies aBeltrami equation gz = µgz, see [Ken79]. This differential equation was used byGauss to prove the local existence of isothermal coordinates on a surface with ana-lytic Riemannian metric, in particular, Gauss proved the existence and uniquenessof a solution in the case that µ is real analytic. Ahlfors noticed a close relationbetween the Beltrami equation and quasiconformal maps, a generalization of con-formal maps. The notion of a quasiconformal map, was introduced by Grötzsch in1928, see [Grö28], and is based on the fact that there is no conformal map from asquare to a rectangle, not square, which maps vertices on vertices. Grötzsch asks forthe most nearly conformal map of this kind. This calls for a measure of approximateconformality and, in supplying such a measure, Grötzsch took the first step towardthe creation of the theory of quasiconformal maps. Roughly speaking, a quasiconfor-mal map is an homeomorphism between complex domains with positive Jacobianthat takes infinitesimal circles into infinitesimal ellipses with eccentricity uniformlybounded. It was pointed by Sa Earp and Toubiana that strengthening the ellipticcondition on the Weingarten relation, the Gauss map is a quasiconformal map. Weprove this assertion and use it together with the Mori theorem, see [Mor56], tocontrol the behavior of the Gauss map at infinity. This allows us to use the Ken-motsu representation to describe the growth of the ends in Cartesian coordinates,obtaining a kind of regularity at infinity for the ends in the sense of Schoen.

Finally, in order to establish that an embedded ESWMT-surface with finite to-tal second fundamental form and two ends is rotationally symmetric, we have tointroduce the Aleksandrov reflection method for non-compact surfaces. In 1956Aleksandrov published one of the most renowned characterization of the sphere,known as the Aleksandrov theorem. This theorem establishes that spheres are thethe only closed connected embedded surfaces in R3 with constant (non-zero) meancurvature. However, more significant than the theorem could be the procedureintroduced by Aleksandrov proving it, called in the literature as the Aleksandrovreflection method, based on the maximum principle. This method was adapted byKorevaar et al. in [Kor+89], using tilted planes, to deal with non-compact surfaces,specifically they proved that a complete connected properly embedded constantmean curvature surface in R3 with two ends is rotationally symmetric; namely, aDelaunay surface. In fact, we will use the extension presented by Aledo et al., in[Ale+10], for Special Weingarten surfaces of elliptic type.

Summarizing, we have several background to set our main theorem, a Schoentype theorem for embedded ESWMT-surfaces in the Euclidean space R3 with finitetotal second fundamental form and two ends.

xx

We are going to describe the results obtained in this thesis.

In Chapter 2, we obtain a representation formula for a SW-surface immersed inR3, using a special Codazzi pair, that depends of the Gauss map, g, of the immersionand the function associated to the surface, Theorem 2.16. Furthermore, we showthat the Gauss map satisfies a second order differential equation which is a completeintegrability condition for this representation formula:

Theorem 2.16. Let M be an oriented SW-surface, then the Gauss map g satisfies

2

c− aH

(gzz − 2gzgz

g

1 + |g|2)

= −(a+ 1

c− aH

)

zgz +

(a− 1

c− aH

)

zgz,

where the functions a and c are given by (2.13) and (2.14) respectively.

The first result in Chapter 3 is called the growth lemma, Lemma 3.10. It esta-blishes that an embedded end of an ESWMT-surface in R3 with finite total secondfundamental form cannot grow infinitely in opposite directions. Formally speaking,we prove that if the Gauss map at infinity is vertical and the end is the graph ofa function φ defined outside some compact set in a horizontal plane, then after avertical translation of the end, either φ ≤ 0 or φ ≥ 0.

Lemma 3.10 (Growth lemma). Let E be an end of an immersed ESWMT-surface.

Suppose there exists R > 0 such that E is obtained as the graph of a C2 class function

φ defined in R2 \ B(R). Suppose also that |∇φ| → 0 as r → ∞, where r is the

distance from an arbitrary point to the origin. Then at least one of the quantities

inf|(x,y)|≥R

φ(x, y) or sup|(x,y)|≥R

φ(x, y) is finite.

As a consequence of the growth lemma we have an obstruction for non-planarone ended ESWMT-surfaces with finite total second fundamental form in R3. Thisis an evidence about the premise that the theory of ESWMT-surfaces is very similarto the theory of minimal surfaces, even more if we note that there is not suchobstruction for harmonic surfaces with finite total curvature, see [Con+15].

Theorem 3.11. Let M be a complete connected ESWMT-surface in R3 with finite total

second fundamental and one embedded end. Suppose that the function f associated

to M is non-negative, Lipschitz at 0 and satisfies lim inft→0+

4t(f ′(t)

)2< 1. Then M is a

plane.

The main result of Chapter 3, Theorem 3.21, is a full representation of the heightfunction in the growth lemma and can be interpreted as a kind of regularity atinfinity for ends of ESWMT-surfaces with finite total second fundamental form.

Theorem 3.21. Let M be a complete ESWMT-surface in R3 with finite total second

fundamental form and embedded ends. Suppose that the function f associated to M

is non-negative and Lipschitz. Then each end Ei is the graph of a function φi over the

exterior of a bounded region in some plane Πi. Moreover, if x1, x2 are the coordinates

xxi

in Πi, then the function φi have the following asymptotic behaviour for r =√x2

1 + x22

large

φi(x1, x2) = β log r + a0 +a1x1

r2/ǫ+a2x2

r2/ǫ+O

(r−2/ǫ

)+ o (log r) ,

where ǫ, β, a0, a1, a2 are real constants depending on i and 0 < ǫ ≤ 1.

The main result of this work can be found in Chapter 4. It establishes that therotational examples of ESWMT-surfaces, characterized by Sa Earp and Toubiana,are the only ones embedded with finite total second fundamental form and twoends. This result can be understood as a Schoen type theorem for ESWMT-surfaces,it reads as follows:

Theorem 4.4 (Main Theorem). Let M be a complete connected ESWMT-surface em-

bedded in R3 with finite total second fundamental form and two ends. Suppose that

the function f associated to M is non-negative and Lipschitz. Then M must be rota-

tionally symmetric. In fact, there exists τ > 0 such that M is the surface of revolution

Mτ determined by Sa Earp and Toubiana in [ST95].

In the subsection 3.4.3 we will show that the Lipschitz condition for the functionf in the Main Theorem can be weakened to the condition lim sup

t→0+

4t(f ′(t)

)2< 1.

As a direct consequence of the last theorem, we can recover the original Schoentheorem.

Corollary 4.6. Let M be a complete connected embedded minimal surface in R3 with

finite total curvature and two ends. Then M is a catenoid.

Thesis Structure

This document is organized in four chapters that develop, step by step, all thetools required to prove Theorem 4.4.

Chapter 1

The first chapter is devoted to the introductory material that we will use in theremaining chapters. It starts with a brief presentation of the first and second fun-damental forms of an immersion. Also, here we present an alternative for theWeierstrass representation used in the Schoen theorem, namely the Kenmotsu re-presentation. We explore other types of finite total curvature surfaces based on thework of White in [Whi87]. We recall the Aleksandrov reflection method for ESWMT-surfaces. Finally, we introduce a Codazzi pair, which is a pair of bilinear forms, oneof them positively defined, satisfying the Codazzi equation. This is a concept thatgeneralizes the first and second fundamental forms.

Chapter 2

xxii

In order to show the similarities between minimal surfaces and ESWMT-surfaces,we prove that an ESWMT-surface satisfies some basic properties that come fromminimal surface theory, as the convex hull property, for instance. We summarize theresults of Sa Earp and Toubiana in [ST95] that characterize the rotational examplesof ESWMT-surfaces. Besides, following the work of Kenmotsu in [Ken79] and usingCodazzi pairs we give a representation formula for SW-surfaces, which is based onthe Weingarten relation.

Chapter 3

Here, we use the finite total second fundamental form hypothesis and the Ken-motsu representation to prove that the height function, in a conformal parameter,associated to an end of an ESWMT-surface is of logarithmic order.

Chapter 4

The final chapter uses the Aleksandrov reflection method to show that an embed-ded ESWMT-surface with finite total second fundamental form and two ends mustbe rotational. This fact joined with the results of Sa Earp and Toubiana give us aSchoen type theorem for ESWMT-surfaces.

xxiii

1Preliminaries

„Mathematicians do not study objects,

but relations between objects.

— Henri Poincaré

This chapter contains basic concepts that we will use along this thesis. At afirst glance, the material might seem quite independent, however, throughout thedocument these contents will relate among themselves to shape the main result ofthis work. Readers familiarized with the topics presented here can skip the entirechapter.

This chapter is based on [Ken79; Mil80; Whi87; Kor+89; PS07; Esp08; Ale+10].

Along this document we will consider M as a connected oriented surface en-dowed with a Riemannian metric, which is isometrically immersed in the Euclideanspace R3. Since every immersion is locally embedded, for any point at M thereexists a neighborhood in M such that its image by the immersion is a regular sur-face contained in R3. This fact allows us, in order to simplify the notation, tolocally identify M with its image in the Euclidean space and every tangent vectorof M with its image by the differential of the immersion. That is, if f : M → R3 isan isometric immersion, then for any p ∈ M there exists a neighborhood U ⊂ M ofp such that f(U) ⊂ R3 is a regular surface and we will identify p with f(p), U withf(U) and v ∈ TpM with dfp(v) ∈ Tf(p)R

3 ≃ R3. Using these identifications and thescalar product of R3 we have the following decomposition at any point p ∈ M

R3 ≃ TpR

3 = TpM ⊕NpM,

where NpM is the orthogonal complement of TpM in R3. In this way we can nowconsider the normal bundle NM of M , whose fiber at p is NpM . We will denote byX(M) the set of tangent vector fields on M , that is, X ∈ X(M) if X : M → TM isa smooth map; in the same way X(M)⊥ will be the set of normal vector fields onM . Since M is oriented, there exists a unit normal vector field N ∈ X(M)⊥. GivenX,Y ∈ X(M) and X, Y local extensions to R3 of these fields, we have that theLevi-Civita connection of R3 and M , denoted by ∇ and ∇ respectively, are relatedby the equation

∇XY =(∇X Y

)T.

1

1.1 Fundamental forms of an immersion

We will consider onM two bilinear forms, denoted by I and II, induced naturally.We are talking about the first and second fundamental forms of M that are definedby the Euclidean metric induced on the surface, and the bilinear form associated tothe Weingarten endomorphism, respectively. We will formalize these ideas and fixsome notations.

We will denote by D(M) the set of C∞ maps on M valued in R. Let N ∈ X(M)⊥

be a unit normal vector field. We define I, II : X(M) × X(M) → D(M) forX,Y ∈ X(M) as

• I(X,Y ) = 〈X,Y 〉, where 〈, 〉 is the scalar product of R3.

• II(X,Y ) = 〈−∇XN,Y 〉.

The bilinear forms I and II are called first and second fundamental forms of M ,respectively.

On the one hand, note that the first fundamental form is definite, that is, thequadratic form associated to I is definite, actually positive definite. On the otherhand, it is easy to show that ∇XN , where X ∈ X(M), is a tangent field on M andthat the endomorphism S of X(M) defined by SX = −∇XN is self-adjoint; thismeans that the second fundamental form is symmetric. The map S is known asthe Weingarten endomorphism or shape operator. The second fundamental formand the Weingarten endomorphism are mutually determined, that is, one of themdetermines in a unique way the other one by the equation II(X,Y ) = I(SX, Y ) forall X,Y ∈ X(M).

The first and second fundamental forms are related by the Codazzi equation,

∇XSY − ∇Y SX = S[X,Y ], (1.1)

where X,Y ∈ X(M) and [X,Y ] is the bracket of the tangent fields X and Y . TheCodazzi equation is a direct consequence of the flatness of the Euclidean space,that is, the fact that the curvature tensor, R, of R3 vanishes identically, where R isdefined as R(X,Y,Z) = ∇X∇Y Z − ∇Y ∇XZ − ∇[X,Y ]Z, where X,Y,Z are vectorfields in R3.

1.1.1 Curvatures of an immersion

Let p ∈ M and v ∈ TpM . Since the value ∇XN(p) depends only on the valueX(p), we can consider the self-adjoint linear transformation Sp : TpM → TpM

defined by Sp(v) = −∇XN(p), where X ∈ X(M) is any tangent vector field suchthat X(p) = v. This observation allows us to write

Sp(v) = −∇vN = −dNp(v).

2 Chapter 1 Preliminaries

By the symmetry of Sp, there exists an orthonormal basis v1, v2 of TpM andk1(p), k2(p) ∈ R such that

Sp(v1) = k1(p)v1 and Sp(v2) = k2(p)v2.

The principal curvatures of M at p are defined as the eigenvalues k1(p), k2(p) of Sp

and can be obtained as the maximum and the minimum of the second fundamentalform at p restricted to the unit circle of TpM , see [do 76].

A point p ∈ M is called umbilical if k1(p) = k2(p). A surface such that all pointsare umbilical is named totally umbilical, or simply, umbilical. An interested factabout totally umbilical surfaces in R3 is that they are, even locally, spheres andplanes, see [do 76; Daj90].

The Gaussian curvature, K, and the mean curvature, H, of M at p are defined asthe determinant and the normalized trace of Sp, respectively. That is, in terms ofthe principal curvatures

K(p) = k1(p)k2(p) and H(p) =1

2

(k1(p) + k2(p)

).

Remark 1.1. One of the most remarkable theorems of Gauss says that for an immersed

surface in R3 the Gaussian curvature, K, is invariant under local isometries. Actually

this quantity depends only on the metric structure of the surface, that is, the first

fundamental form. In other words, the “extrinsic” Gaussian curvature is an intrinsic

invariant of the surface.

It is known that the determinant and the trace of a linear map are invariants ofthe map, i.e., they do not dependent on the choice of the basis. This fact gives usan alternative way to compute the Gaussian and mean curvature. Considering thematrices associated to the first and second fundamental forms in any basis of TpM ,and denoting this matrices in the same way, I(p) and II(p) respectively, it is easy toshow that the Gaussian and mean curvature are determined by

K(p) =det(II(p))

det(I(p))and H(p) =

1

2

(tr(I(p)−1II(p))

). (1.2)

A straightforward computation shows that we can recover the principal curva-tures from the mean and Gaussian curvatures,

k1(p) = H(p) +√H(p)2 −K(p) and k2(p) = H(p) −

√H(p)2 −K(p). (1.3)

Note thatq(p) = H(p)2 −K(p) =

1

4

(k1(p) − k2(p)

)2, (1.4)

which means that q(p) = 0 if, and only if, p ∈ M is an umbilical point. The valueq(p) is called the skew curvature of M at p.

1.1 Fundamental forms of an immersion 3

The norm of the second fundamental form at p ∈ M is defined using the principalcurvatures as

|II(p)| =√k1(p)2 + k2(p)2. (1.5)

From the above definition we have that

|II|2 = 4H2 − 2K. (1.6)

1.1.2 Fundamental forms in coordinates

Let (U,x = (x, y)) be a local chart of M . The fundamental forms (I, II) can bewritten using these coordinates as

I = Edx2 + 2Fdxdy +Gdy2 and II = edx2 + 2fdxdy + gdy2,

where E,F,G, e, f, g are smooth functions defined on U . Using these functions and(1.2) we can compute the Gaussian and mean curvature of the immersion as

K =eg − f2

EG− F 2and H =

Eg − 2Ff +Ge

2(EG − F 2).

Now consider the complex parameters (z, z) on U , where z = x+ iy; then defining∂z = 1

2 (∂x − i∂y) and ∂z = 12 (∂x + i∂y), M can be viewed as a Riemann surface,

and CTM as the complexification of the tangent bundle of M , generated by the set∂z, ∂z. We rewrite the first and second fundamental forms as

I = Pdz2 + 2λ|dz|2 + P dz2 and II = Qdz2 + 2ρ|dz|2 + Qdz2,

where P, λ,Q, ρ are complex functions. In these coordinates, the curvatures can bedetermined as

K =|Q|2 − ρ2

|P |2 − λ2and H =

PQ− 2λρ+ PQ

2(|P |2 − λ2).

The equations that relate the corresponding functions in the (x, y)−coordinatesand the (z, z)−coordinates are:

P =1

4(E −G− 2iF ), Q =

1

4(e− g − 2if),

λ =1

2(E +G), ρ =

1

2(e+ g).

Note that λ and ρ are real functions.


Isothermal (conformal) parameters

We say that (x, y) are isothermal parameters for the first fundamental form I, ifE = G > 0 and F = 0, thus I = E(dx2 + dy2). We also say that the complex para-meter z = x+ iy is a conformal parameter for I if P = 0, so I = 2λ|dz|2. Thereby,from the equations presented above, the parameter z = x + iy is a conformal pa-rameter for I if, and only if, (x, y) are isothermal parameters for I.

It is well known that, when the surface is regular enough, at any point of thesurface there exists a local isothermal chart, see for instance [Che55a]. Also, if thesurface is oriented, a positive oriented atlas of isothermal charts gives to the surfacea structure of a Riemann surface.

Lemma 1.2. Let I, II be the first and second fundamental forms, respectively, and z

be a conformal parameter for I. Then

I = 2λ|dz|2 and II = Qdz2 + 2λH|dz|2 + Qdz2, (1.7)

and the following equations hold,

K = H2 − |Q|2λ2

and |Q|2 = qλ2. (1.8)

The equation on the right in (1.8) says that p ∈ M is umbilical if, and only if,Q(p) = 0. In particular, if Q vanishes identically on M , then M is totally umbilicalin R3.

The Hopf theorem

Let us observe that, by (1.7), the Weingarten endomorphism is determined by theequalities,

S∂z = H∂z +Q

λ∂z and S∂z =

Q

λ∂z +H∂z. (1.9)

From (1.9), it is easy to conclude that the Codazzi equation (1.1) is equivalent to

Qz = λHz, (1.10)

which implies that the mean curvature, H, is constant if, and only if, the quadraticdifferential form Qdz2, which does not depend on the chosen parameter z, is holo-morphic. Hence, if H is a constant and M is a topological sphere, the differentialQdz2 must vanish identically on M . We have proven the celebrated Hopf theorem,which asserts

Theorem 1.3 (Hopf Theorem). Any topological sphere immersed in the Euclidean

space R3 with non-zero constant mean curvature must be a round sphere.

1.1 Fundamental forms of an immersion 5

The quadratic differential form Qdz2 is called the Hopf differential, see [Hop83;AR04; Esp08; Ale+10].

1.2 The Gauss map and the representation of surfaces

in the Euclidean space

Given a unit normal vector field N ∈ X(M)⊥, called Gauss map, we can considerN as a function into the unit sphere of R3, and think this sphere as the standardRiemann sphere C ∪ ∞. From this point of view, the Gauss map can be rewrittenas the composition of the usual stereographic projection with N , we will denotethis function by g : M → C∪ ∞, which will be also called the Gauss map of M .

The Gauss map of an immersion in R3 has been of great interest in DifferentialGeometry. Several authors have shown that this map encodes a large amount ofinformation about the surface, and, in some cases it can determine the existenceand uniqueness of an immersion with a given conformal structure, see for example[Oss69; AE75; Ken79; HO83; HO85]. There are two well-known representations ofa surface based on the Gauss map: the Weierstrass-Enneper representation and theKenmotsu representation. However, there are other representations, as for exam-ple the one obtained by Gálvez and Martínez in [GM00] for a surface with positiveGaussian curvature using the conformal structure given by the second fundamen-tal form. In the following we will introduce briefly the Weierstrass-Enneper andKenmotsu representations.

1.2.1 The Weierstrass-Enneper representation

An oriented surface M immersed in R3 is called minimal if the mean curvaturevanishes everywhere. Minimal surfaces are one of the most studied topic in Diffe-rential Geometry; they are intimately related with harmonic and complex functions.The Weierstrass-Enneper representation establishes this relation and it says that anyminimal surface may be represented by an holomorphic function and a meromor-phic function, called Weierstrass data.

Let (U,x = (x, y)) be a local chart of M , where (x, y) are isothermal parametersfor the first fundamental form, and consider the complex parameter z = x + iy,which is conformal. By Lemma 1.2 we know that I = 2λ|dz|2. It is well known that∆x = 2λHN , where ∆ is the Laplace-Beltrami operator (see [do 76]). This allowsus to conclude that M is minimal in R3 if, and only if, the coordinate functions areharmonic.

In order to establish the relation between minimal surfaces and analytic functionsof a complex variable we define

(φ1, φ2, φ3) = Φ = ∂zx and ((Φ,Φ)) = φ21 + φ2

2 + φ23.


A straightforward computation shows that z is a conformal parameter if, and onlyif, ((Φ,Φ)) = 0, and ∆x = 0 if, and only if, ∂zΦ = 0, see [Oss69]. Note that thecondition ∂zΦ = 0 says that all the components of Φ are holomorphic, and in thiscase we will say that Φ is holomorphic.

We can recover the immersion x, on simply connected domains, from Φ by theCauchy theorem as

x(z) = ℜ∫ z

z0

Φ(w) dw,

then, in order to get locally minimal surfaces, we focus on holomorphic solutions ofthe equation ((Φ,Φ)) = 0. The next lemma characterizes those solutions.

Lemma 1.4 (Osserman, [Oss69]). Let U ⊂ C be a domain and g : U → C a

meromorphic function. Let f : U → C be a holomorphic function having the property

that each pole of g of order m is a zero of order at least 2m. Then,

Φ =

(1

2f(1 − g2),

i

2f(1 + g2), fg

)(1.11)

is holomorphic and satisfies ((Φ,Φ)) = 0. Conversely, every holomorphic Φ satisfying

((Φ,Φ)) = 0 may be represented in the form (1.11), except for Φ = (φ,−iφ, 0).

The lemma above implies the next theorem which gives us a local representationof minimal surfaces in R3.

Theorem 1.5 (Weierstrass-Enneper representation). Every minimal surface immersed

in R3 can be locally represented in the form

x(z) = ℜ∫ z

0Φ(ω) dω

+ c (1.12)

where c is a constant in R3, Φ is defined by (1.11), the functions (f, g) having the

properties stated in Lemma 1.4, the domain U being either the unit disk or the entire

complex plane, and the integral being taken along an arbitrary path from the origin

to the point z. The surface x(U) will be regular if, and only if, f vanishes only at the

poles of g, and the order of each zero is exactly twice the order of this point as pole of

g.

Remark 1.6. It is easy to check that if an immersed minimal surface is represented by

(1.12), the Gauss map is exactly the meromorphic function g.

As an example of this representation we can obtain the catenoid, see Figure 1.1,for the functions f, g : C \ 0 → C, where f(z) = e−z and g(z) = ez. Then by(1.11), we get

Φ(ω) =

(1

2eω(1 − e2ω),

i

2eω(1 + e2ω), 1

).

1.2 The Gauss map and the representation of surfaces in R3 7

The immersion is achieved as

x(z) = ℜ∫ z

0

(e−ω − eω

2,i(e−ω + eω)

2, 1

)

= ℜ∫ z

0(− sinh(ω), i cosh(ω), 1)

= ℜ (− cosh(z), i sinh(z), z)

x(x, y) = (− cosh(x) cos(y),− cosh(x) sin(y), y) .

It is well known that the catenoid is the only non-planar minimal surface that isrotationally symmetric. Besides, it is properly embedded in R3, and conformallythe punctured plane, therefore parabolic.

Figure 1.1: Catenoid.

1.2.2 The Kenmotsu representation

A natural question is: can we generalize the Weierstrass-Enneper representationfor any other kind of surface? The answer is yes, we can have a representation quitesimilar to the Weierstrass-Enneper one for any immersed surface in R3, however,the complex functions involved in the representation may not be holomorphic. Oneof this kind of representations was presented by Kenmotsu in [Ken79].

Let M be an oriented surface immersed in R3 and z = x + iy be a conformalparameter for the first fundamental form, I. We consider the smooth unit normalvector field N defined by N = 1

E (∂x ∧ ∂y), where ∧ is the exterior product inR3. The vector field Nz = ∂z(N) is tangent to the surface. The following lemmaestablishes, using Lemma 1.2, the representation of this tangent vector field in thebasis ∂z , ∂z.

Lemma 1.7. −Nz = H∂z + Qλ ∂z.

Proof. Since −Nz is a tangent vector field, there exist complex functions c1 and c2

such that −Nz = c1∂z + c2∂z. Then I(−Nz, ∂z) = c2λ and I(−Nz, ∂z) = c1λ. SinceI(−Nz, ∂z) = II(∂z, ∂z) = Q and I(−Nz, ∂z) = II(∂z, ∂z) = λH, we obtain theresult.


The coordinates (x, y) are isothermal, then from the definition of N we have thatN ∧ ∂x = ∂y and N ∧ ∂y = −∂x. Thus,

N ∧ ∂z = N ∧(

1

2(∂x − i∂y)

)=

1

2(∂y + i∂x) = i∂z and N ∧ ∂z = −i∂z. (1.13)

This computations will allow us to generate the tangent bundle using the informa-tion obtained from the unit normal vector field N and its derivative Nz, at any pointoutside of umbilical points and minimal points. Specifically, let us verify that thevectors Nz and iN ∧ Nz are linearly independent whenever HQ does not vanish.From Lemma 1.7 and (1.13) we get

iN ∧Nz = iN ∧(

−H∂z − Q

λ∂z

)= H∂z − Q

λ∂z. (1.14)

Then, it is very easy to see that the vectors make a tangent basis under the conditionHQ 6= 0.

Now we can recover the natural basis induced by the immersion at any point withnon-zero mean curvature as we show in the next Lemma.

Lemma 1.8. ∂z = 12H (iN ∧Nz −Nz).

Proof. It follows immediately from (1.14) and Lemma 1.7.

As we mentioned above, we are considering the stereographic projection of theunit normal N = (N1, N2, N3) from the north pole, that is,

g(z) =N1(z)

1 −N3(z)+ i

N2(z)

1 −N3(z).

We will call this function, again, the Gauss map of the surface M . Hence,

N =1

1 + gg(g + g,−i (g − g) ,−1 + gg) ,

and Nz = gz ξ + gzξ, where ξ is the vector field defined as

ξ =1

(1 + gg)2

(1 − g2, i(1 + g2), 2g

).

A straightforward computation shows that iN ∧ ξ = −ξ, and then by conjugationiN ∧ ξ = ξ. This helps us to obtain

iN ∧Nz = iN ∧(gz ξ + gzξ

)= gz ξ − gzξ,

which means that we can obtain ∂z from g, gz and gz. Specifically,

Theorem 1.9. ∂z = − 1H gzξ.

1.2 The Gauss map and the representation of surfaces in R3 9

Proof. Directly from Lemmas 1.8 and the above equations.

When the surface is minimal, it is well known that the Gauss map is a holomor-phic function. In general, it might be not true that the Gauss map is holomorphic,however, this map satisfies a Beltrami equation.

Theorem 1.10 (Beltrami equation). The Gauss map g of an immersed surface M in

R3 must satisfy a Beltrami equation Hgz = Qλ gz .

Proof. Lemma 1.7 says that −Nz = H∂z + Qλ ∂z, and then by conjugation we have

that −Nz = Qλ ∂z +H∂z. Note that Nz , Nz is also a tangent basis, then we recover

∂z in this basis as

∂z = − 1

K

(HNz − Q

λNz

). (1.15)

From Nz = gz ξ + gzξ, we obtain by conjugation Nz = gzξ + gz ξ. Then, using thisexpressions and Theorem 1.9 in (1.15) we get,

K

Hgzξ = −K∂z =

(Hgz − Q

λgz

)ξ +

(Hgz − Q

λgz

)ξ. (1.16)

Since the vector fields ξ and ξ are linearly independent, we conclude that the coeffi-cient of ξ in (1.16) vanishes, that is, Hgz − Q

λ gz = 0, as desired.

Note that when the immersion is minimal, the Beltrami equation implies thatgz = 0, which means that the Gauss map is a holomorphic function; this fact waspointed by Osserman in [Oss69]. Hence, the Beltrami equation can be seen as ageneralization of the Cauchy-Riemann equations for g in the minimal case.

The Gauss map also satisfies a second order differential equation, which is acomplete integrability condition for the equation in Theorem 1.9. This differentialequation is presented in the next theorem.

Theorem 1.11. The Gauss map g of an immersed surface M in R3 must satisfy

H

(gzz − 2g

1 + gggzgz

)= Hzgz. (1.17)

The proof of the last theorem can be found in [Ken79], and it is mainly a conse-quence of the Codazzi equation (1.1) and the Theorems 1.9 and 1.10.

Now we have all elements to recover the immersion from the Gauss map and thenon-zero mean curvature. This means that we can have a representation formulafor immersed surfaces prescribing the mean curvature and the Gauss map. Suchrepresentation, presented bellow, can be seen as a generalization of the Weierstrass-Enneper formula for minimal surfaces. This result was proved by Kenmotsu in[Ken79].


Theorem 1.12. Let M be a simply connected 2-dimensional smooth manifold and

H : M → R be a non-zero C1−function. Let g : M → C ∪ ∞ be a smooth

map satisfying (1.17) for H. Then there exists a branched immersion x : M → R3

with Gauss map g and mean curvature H. Moreover, the immersion is unique up to

similarity transformations of R3 and it can be recovered using the equation

x = ℜ∫

− 1

Hgzξ dz

+ c, (1.18)

where c is a constant in R3 and the integrals are taken along a path from a fixed point

to a variable point. Moreover, if gz 6= 0, then x(M) is a regular surface.

Proof. This follows from Theorem 1.9 and Theorem 1.11.

1.3 Finite total curvature

This section is based on the work of Huber and White, especially on [Hub57;Hub66] and [Whi87]. The aim is to know the relation between the topology andthe conformal structure of a non-compact complete surface and the finiteness ofsome geometric quantities. The first result in this direction is due to Cohn-Vossenin [CV33], where he proved that for a finitely connected and complete Riemanniansurface, the integral of the Gaussian curvature is finite and at most 2π times theEuler characteristic of the surface. This integrability of the Gaussian curvature isknown in the literature as finite total curvature, see for example [CV33; Fin65;HT92; LT91; Oss69; Shi85].

Definition 1.13. An immersed surface M in R3 is said to have finite total curvature

if ∣∣∣∣∫

MK dA

∣∣∣∣ < +∞. (1.19)

Huber in [Hub57] proved that finite total curvature implies finite connectivity,which means that the hypothesis finitely connected in the Cohn-Vossen theoremcan be replaced by connected. His theorem reads as

Theorem 1.14 (Huber, [Hub57]). If M has finite total curvature, then M is of finite

topological type, moreover M is conformally equivalent to M \ p1, · · · , pk, where M

is a compact Riemann surface.

Here we will work with the integral of the squared norm of the second fundamen-tal form, recall (1.5) and (1.6).

Definition 1.15. An immersed surface M in R3 is said to have finite total second

fundamental form if ∫

M|II|2 dA < +∞. (1.20)

1.3 Finite total curvature 11

This name is taken from [CM99], however, this concept is also usually named asfinite total curvature, see for example [Whi87; Che94; MŠ95; CM11; AL13]. Weprefer to avoid confusions and use those names separately.

Note that a minimal surface in R3 has finite total second fundamental form if,and only if, has finite total curvature, since

|II|2 = k21 + k2

2 = (k1 + k2)2 − 2k1k2 = 4H2 − 2K = −2K.

This equivalence is also true for quasiconformal harmonic immersions in R3, see[AL13].

In general, finite total second fundamental form is stronger than finite total cur-vature; a fact proved by White in [Whi87].

Theorem 1.16 (White, [Whi87]). If M has finite total second fundamental form,

then M has finite total curvature, indeed∫

M K dA = 2πm for some integer m.

In 1964, Osserman in [Oss64] showed that, for minimal surfaces in R3, the totalcurvature is related to the distribution of the normals. Specifically, he proved thatif a complete minimal surface has finite total curvature, then the surface is confor-mally equivalent to a compact Riemann surface minus a finite number of points, andthe Gauss map extends to a meromorphic function at those points. It is possible toobtain an Osserman’s type theorem removing the hypothesis of being minimal. Thetopological conclusion can be deduced without the minimal hypothesis, Theorem1.14; the second implication was obtained by White in [Whi87] assuming finite to-tal second fundamental form and non-positive Gaussian curvature, in this case theextension is, a priori, only continuous.

Theorem 1.17 (White, [Whi87]). Suppose M has finite total second fundamental

form, in particular, M ≈ M \p1, · · · , pk as in Theorem 1.14. Let Ui be a (punctured)

neighborhood of pi in M . If K does not change sign in Ui, then the Gauss map extends

continuously to pi, i = 1, . . . , k, and M is properly immersed near pi.

1.4 Maximum principles and geometric applications

The maximum principle is a classical result on the theory of second order ellipticpartial differential equations, it leads us to know information about solutions ofdifferential equations and inequalities without any explicit knowledge of the solu-tions themselves. Initially, it was proposed separately by Gauss and Earnshaw forharmonic and subharmonic functions (1839) and, from then, several mathemati-cians have generalized those results as, for example, Paraf (1892), Moutard (1894),Picard (1905), Picone (1927) and Hopf (1952).

Here we are interested in the application of the maximum principle in geometry;it becomes a powerful tool since surfaces are locally graphs of functions defined in a


domain of the tangent plane and elliptic partial differential equations arise naturallyin the study of the curvatures of the surface. This was the idea of Aleksandrov[Ale56] for proving that, a closed connected embedded constant (non-zero) mean

curvature surface in the Euclidean three space must be a round sphere; theorem knownnowadays as the Aleksandrov theorem.

1.4.1 The Hopf Maximum principle

The maximum principle we will present here is due to Hopf, see [Hop27; Hop52].For a very detailed proof of it, we refer the reader to [PS07]. It is based on theobservation that if a function of class C2 on a domain attains a maximum value atcertain point, then at this point the gradient of the function must vanish and theHessian matrix must be non-positive definite.

Let Ω ⊂ Rn be a domain and set L a second order partial differential operator ofthe form

L =∑

i,j

aij(x)∂2xixj

+∑

i

bi(x)∂xi, (1.21)

for given real continuous functions aij and bi defined on Ω satisfying aij = aji.

The partial differential operator L is called uniformly elliptic if there exists apositive constant δ such that

∑

i,j

aij(x)yiyj ≥ δ|y|2, (1.22)

for any x ∈ Ω and y = (y1, . . . , yn) ∈ Rn. We say that L is locally uniformly ellipticif (1.22) holds on any compact subset of Ω, where the constant δ is now dependingof the compact subset.

Note that the ellipticity of L means that for each point x ∈ Ω, the symmetricmatrix (aij(x)) is positive definite, with smallest eigenvalue greater than or equalto δ.

Theorem 1.18 (Maximum principle). Let u be a C2(Ω) function which satisfies the

differential inequality Lu ≥ 0 in the domain Ω, where L is a locally uniformly elliptic

operator. If u takes a maximum value C in Ω, then u ≡ C in Ω.

The Interior maximum principle does not apply when the function u attains itsmaximum value at the boundary of the domain Ω. The following theorem treatsthis case.

Theorem 1.19 (Hopf Lemma). Let u be a C2(Ω) function continuous in ∂Ω which

satisfies the differential inequality Lu ≥ 0 in the domain Ω, where L is a locally

uniformly elliptic operator. Suppose that Ω satisfies the interior sphere condition and

let x0 ∈ ∂Ω such that

(i) u is of class C1 in x0.

1.4 Maximum principles and geometric applications 13

(ii) u(x0) ≥ u(x) for all x ∈ Ω.

(iii) ∂νu(x0) ≥ 0, where ν is the inward normal direction of ∂Ω.

Then u is constant.

1.4.2 Comparison principle for surfaces

In order to apply Theorems 1.18 and 1.19 in a geometric context, we will give thedefinitions of interior tangency point and boundary tangency point that will help usto situate the local description of surfaces.

Definition 1.20. Let M1 and M2 be immersed surfaces in R3 and p ∈ M1 ∩ M2. The

point p is called a tangent point if TpM1 = TpM2, and either p is an interior point of

both surfaces or p ∈ ∂M1 ∩ ∂M2 and the interior conormals vectors of ∂M1 and ∂M2

coincide at p.

Given a surfaceM immersed in R3 and p ∈ M , we can describe locally the surfaceM in a neighborhood of p as a graph of a smooth function defined on its tangentplane at p, thinking of this linear space as an affine plane of R3 passing throughp. That is, there exists a neighborhood V ⊂ M of p, a domain Ω ⊂ TpM and afunction u : Ω → R of class C2 on Ω such that

M ∩ V = Gra (u) = q + u(q)N(p) : q ∈ Ω,

where N is a unit normal vector field defined on M . Note that u(p) = 0. Whenthe surface M has no empty boundary and p ∈ ∂M , the function u is defined on Ω,p ∈ ∂Ω and u(∂Ω) = ∂M ∩ V .

Note that when the surfaces M1 and M2 have a common tangency point p (at theinterior or at the boundary), then locally those surfaces can be obtained as graphs(over the same plane) of functions u1 and u2 respectively, defined on the same do-main (Ω or Ω according if either p is an interior or a boundary point respectively).

Let us denote by M1 ≥p M2 if M1 and M2 have the point p as a common tangencypoint and u1 ≥ u2 in a neighborhood of p in Ω (or Ω), which means geometricallythat locally the surface M1 is above of the surface M2 around p.

Now we state a geometric maximum principle using the functions that locallydescribe the surfaces.

Definition 1.21. Let M1 and M2 be surfaces immersed in R3. We say that the sur-

faces satisfy the maximum principle if, for every common tangent point p ∈ M1 ∩ M2,

the function w = u2 − u1 satisfies a differential inequality Lw ≥ 0 for some locally

uniformly elliptic operator L in the form (1.21) and, in addition, the conditions in the

hypothesis of Theorem 1.19 when the tangent point is at the common boundary of the

surfaces. Here ui is the function such that Mi is locally, around p, represented as the

graph of ui, for i = 1, 2.


The intention of the last definition is to apply Theorem 1.18 and Theorem 1.19thinking just in the geometric fact that two surfaces are touching tangentially eachother. Hence, using the Definition 1.21 and the theorems mentioned we obtain thefollowing geometric comparison principle.

Theorem 1.22 (Geometric comparison principle). Let M1 and M2 be surfaces im-

mersed in R3 satisfying the maximum principle. If M1 ≥p M2 then M1 = M2.

It is common to work with families of surfaces satisfying the maximum principle,see for example [Ale+10; Esp08].

Definition 1.23. Let F be a family of oriented immersed surfaces in R3. We say that

F satisfies the maximum principle if the following properties are fulfilled:

(i) F is invariant under isometries of R3.

(ii) If M ∈ F and M is another surface contained in M , then M ∈ F.

(iii) Any two surfaces in F satisfy the maximum principle, in the sense of Definition

1.21.

As examples of families of surfaces satisfying the maximum principle we wouldlike to mention the family of minimal surfaces and the family of surfaces with non-zero constant mean curvature. In both cases, they achieve the conditions of Defini-tion 1.23 mainly because the equation H = c, where c is a constant (c = 0 for thecase of minimal surfaces), gives a quasilinear elliptic equation when the surface iswritten as a graph. Although these families come from a similar equation they arequite different; for example there are no compact minimal surfaces while the sphereof radius 1

c is the unique compact embedded surface of constant mean curvature c(the Aleksandrov Theorem).

1.4.3 The Aleksandrov reflection method

As we have mentioned, Aleksandrov characterized spheres as the only closedconnected surface embedded in R3 with constant (non-zero) mean curvature. Al-though this is a major theorem, the procedure introduced by Aleksandrov have hada remarkable impact in the development of Differential Geometry. This technique iscalled the Aleksandrov reflection method and it is based in the geometric compari-son principle, Theorem 1.21.

Aleksandrov’s idea, roughly speaking, was to compare the surface with succe-ssive reflections of parts of the surface looking for a possible first tangency point,and finding at this time a plane of symmetry for the original surface. With theaim of applying this method later we are going to present the Aleksandrov reflec-tion method following [Kor+89; Ale+10; Esp08], which is a generalization to non-compact surfaces.


Let M be a connected properly embedded surface in R3, then the surface dividesR3 into two connected components. Let denote by N a unit normal vector fieldglobally defined on M . Set C(N) the component of R3 \M pointed by N . Note that∂C(N) = M .

Consider a fixed plane P ⊂ R3 with normal unit vector n. For t ∈ R we willdenote by Pt the parallel plane to P at signed distance t, that is, Pt = P + tn.Hence, the family Ptt∈R is a foliation of R3 by parallel planes to P. Set Pt− andPt+ the closed half spaces, lower and upper respectively, from the plane Pt, i.e.,

Pt− =⋃

s≤t

Ps and Pt+ =⋃

s≥t

Ps.

For any set G ⊂ R3, let Gt+ be the portion of G above the plane Pt and let G∗t+ be

the reflection of this portion through the plane Pt, then we can write

Gt+ = G ∩ Pt+ and G∗t+ = p+ (t − r)n : p ∈ P, p+ (t+ r)n ∈ G. (1.23)

We define analogously the sets Gt− and G∗t− .

Given an open set W ⊂ C(N) consider the portion S of the surface M obtainedby S = ∂W ∩ M . If the set St+ is not empty and S∗

t+ ⊂ W we write S∗t+ ≥ St− ,

which means that the reflection of the set St+ thought the plane Pt is above the setSt− , considering the height from the plane P.

When S is not empty, we will define a function Λ1, called Aleksandrov functionassociated to S.

We start with the definition of the domain of the function Λ1. Given a point p ∈ P,set Lp the perpendicular line to P passing through p, therefore we can parametrizethis line as Lp = p+tn : t ∈ R. Suppose that Lp∩W 6= ∅ but Lp is disjoint from W

for all sufficiently large t, then there exists a value t1(p) such that p+tn /∈ W for anyt > t1(p) and P1(p) = p+ t1(p)n ∈ W . We say that P1(p) is the first point of contactof the set Lp∩W as t decreases from +∞. Now we are going to suppose additionallythat P1(p) ∈ S, then the intersection of Lp and S can be either tangential or elsetransversal. In the first case we will denote P2(p) = P1(p) and t2(p) = t1(p). Whenthe intersection is transversal, we are going to look for the next point where Lp

tries to leave W , if exists, that is, we look for a point P2(p) = p + t2(p)n such thatp+tn ∈ W for any t1(p) ≥ t ≥ t2(p) and P2(p) ∈ S. We say that P2(p) is the secondpoint of contact of the set Lp ∩ W as t decreases from +∞.

Let us denote by D the domain of the Aleksandrov function associated to S anddefine it as the set of points p ∈ P such that either there exists just a first con-tact point or there exist the first and second contact points of the set Lp ∩ W ast decreases from +∞, and those points belongs to S. Geometrically, D containsall points p in the plane P such that the lines Lp, coming from infinity, enter Wthrough S either tangentially or transversally and leaving W also through S.


We define Λ1 : D → −∞ ∪ R the Aleksandrov function associated to S by

Λ1(p) =

t1(p) + t2(p)

2, if there exist P1(p) and P2(p),

−∞, if there exist just P1(p).(1.24)

Note that Λ1(p) is the value such that the reflection of the point P1(p) throughthe plane PΛ1(p) is P2(p).

Definition 1.24. A point p ∈ D is called a local interior maximum for Λ1 if there exists

a neighborhood U of p in P such that for any q ∈ U either q ∈ D, P1(q),P2(q) /∈ ∂S

and Λ1(q) ≤ Λ1(p), or else Lq ∩ W = ∅. Any other local maximum of Λ1 will be called

a local boundary maximum.

Definition 1.25. A first local point of reflection for S with respect to the plane P with

normal n is defined to be a point P2(p) such that p ∈ D and is a local maximum of

Λ1, that is, there exists a neighborhood U of p in P such that Λ1(q) ≤ Λ1(p) for any

q ∈ U ∩ D.

The above definitions are justified through the next lemma which shows that theAleksandrov reflection method can be applied to non-compact surfaces.

Lemma 1.26 (Aleksandrov reflection method). Let F be a family that satisfies the

maximum principle and M ∈ F be a connected surface. Let P be a plane in R3 with

normal n. If, relative to the subsets S ⊂ M and W ⊂ C(N), the Aleksandrov function

Λ1 has a local interior maximum value t0 at p ∈ D, then the plane Pt0 is a plane of

symmetry for M .

Proof. We are going to compare the surface S with the reflection S∗t+0

of St+0

through

the plane Pt0 , keeping in mind the fact that S and S∗t0

are surfaces in F. SinceΛ1(p) = t0 we get that P1(p) reflects to P2(p). The local maximality hypothesisimplies that there exists a neighborhood U ⊂ P of p such that any q ∈ U verifiesthat either q ∈ D and Λ1(q) ≤ Λ1(p), or else Lq ∩ W = ∅. Note that in the secondcase we have that Lq ∩ S = ∅, thus we can assume q ∈ D. Using the definition ofthe Aleksandrov function (1.24) we get

t2(q) ≤ t0 − (t1(q) − t0). (1.25)

Looking at (1.23), we infer that inequality (1.25) means that the reflection of P1(q)

through the plane Πt0 lies above the point P2(q) and then P1(q)∗ ∈ W .

Summarizing, a neighborhood of S∗t+0

containing P2(p) is contained in W . We

have shown that S∗t0

≥P2(p) S or equivalently M∗t0

≥P2(p) M ; in particular P2(p) isan interior tangent point when P1(p) 6= P2(p) and a boundary tangent point whenP1(p) = P2(p). By Theorem 1.22, we conclude that M = M∗

t0, thus Pt0 is a plane

of symmetry of M .


The Aleksandrov reflection method is very useful any time we get a local interiormaximum for the Aleksandrov function. Although the function Λ1 is not conti-nuous in general, it is upper semicontinuous and it will attain its supremum in anycompact domain. This will be an important fact to be used later in Chapter 4.

The next lemma shows that the Aleksandrov function Λ1 is upper semicontinuouswith respect to planes as well as points.

Lemma 1.27. Let S be a closed surface and a parameter ε → 0. Suppose that there

exists a sequence of points pε → p and a sequence of planes Pε → P such that pε ∈ Pε

and p ∈ P. Let Λε1 and Λ1 be the corresponding Aleksandrov functions. If Λε

1(pε) and

Λ1(p) exist, then

lim supε→0

Λε1(pε) ≤ Λ1(p). (1.26)

Proof. Assuming Λε1(pε) exists for ε → 0, then there is a sequence (P1(pε),P2(pε))

of pair of points of S. Since S is closed, a subsequence of this sequence convergesto a pair of points of S, possibly identical, (Q1, Q2). Note that clearly Q1, Q2 ∈ Lp

since Lpε → Lp. We call t1 and t2 the height of these points above P.

The values t1(pε) and t2(pε) converge to the height of Q1 and Q2 above P, respec-tively. By definition of first and second contact points we conclude that the pointsP1(p) and P2(p) must be at least as hight as the points Q1 and Q2 respectively, thatis, t1(p) ≥ t1 and t2(p) ≥ t2. Hence Λ1(p) ≥ t1+t2

2 and (1.26) is verified.

The Aleksandrov theorem

Aleksandrov communicated his theorem and sketched the proof in a lecture givenin Zurich in July 1955. The proof was published in March 1956, see [Ale56], depen-ding on two parts, the first one presented here by Theorem 1.18, Theorem 1.19 andLemma 1.26, and a second one based on the following well-known characterizationof spheres, whose proof can be found in [Hop83].

Lemma 1.28. If a compact surface M is embedded in R3 and has a plane of symmetry

in every direction, then M is a totally umbilical sphere.

Now we can establish a proof for the Aleksandrov Theorem.

Corollary 1.29 (Aleksandrov Theorem). Let F be a family that satisfies the maximum

principle and M ∈ F be a connected compact surface embedded in R3. Then M is a

totally umbilical sphere.

Proof. If M is compact and embedded, then for any plane P the domain of theAleksandrov function Λ1 is a compact set. Therefore, Λ1 reaches its maximum valueat some point p ∈ D. By the Aleksandrov reflection method (Lemma 1.26) a planeparallel to P is a plane of symmetry of M . Since P is an arbitrary plane of R3 weconclude by Lemma 1.28 that M is a totally umbilical sphere.


1.5 Codazzi pairs

In this section we will generalize the concepts introduced in Section 1.1. Herewe are going to consider M as a smooth oriented surface, not necessarily immersedin the Euclidean space or any other manifold, with a pair of real symmetric bi-linear forms prescribed. The outline is to imitate some situations involving the firstand second fundamental forms of an immersion, especially the proof presented ofHopf’s theorem.

We denote by Q(M) the set of symmetric bilinear forms on M and by R(M) thesubset of those forms that are positive definite. A pair (A,B) ∈ R(M) × Q(M) iscalled a fundamental pair.

Once a bilinear form A ∈ R(M) is fixed, there exists a bijective correspon-dence between Q(M) and the set of self-adjoint endomorphisms of X(M) denotedby S(M,A); this correspondence is given by the equation B(X,Y ) = A(SX, Y ),X,Y ∈ X(M), where B ∈ Q(M) and S ∈ S(M,A). We will say that S is theendomorphism associated to the fundamental pair (A,B).

Definition 1.30. Let (A,B) be a fundamental pair on the surface M and S the en-

domorphism associated to this pair. We define the Codazzi tensor associated to the

fundamental pair (A,B) as the map TS : X(M) × X(M) → X(M) given by

TS(X,Y ) = ∇XSY − ∇Y SX − S[X,Y ],

where ∇ is the Levi-Civita connection associated to (M,A).

In the case that the surface M is immersed in R3, the pair (I, II) formed by thefirst and second fundamental forms is, in this context, a fundamental pair on M ,that is, (I, II) ∈ R(M) × Q(M). Moreover, considering the Weingarten endomor-phism S we have that the Codazzi tensor, TS, associated to the pair (I, II) vanishesidentically by the Codazzi equation (1.1). This fact motivates the next definition.

Definition 1.31. Let (A,B) be a fundamental pair on the surface M and S the endo-

morphism associated to this pair. We say that (A,B) is a Codazzi pair if, and only if,

the Codazzi tensor TS associated to this pair vanishes identically.

For a fundamental pair (A,B) we can define real functions that generalize themean and Gaussian curvature, the principal curvatures and the skew curvature ofan immersed surface in the Euclidean space. To do this, we replicate the formulasgiven by (1.2)–(1.4).

Definition 1.32. Let (A,B) be a fundamental pair on the surface M . We define the

mean and extrinsic curvatures of the pair (A,B) as

H(A,B) =1

2tr(A−1B) and Ke(A,B) =

det(B)

det(A).

1.5 Codazzi pairs 19

Also, we define the principal curvatures and the skew curvature of the pair as

• k1(A,B) = H(A,B) −√H(A,B)2 −Ke(A,B),

• k2(A,B) = H(A,B) +√H(A,B)2 −Ke(A,B),

• q(A,B) = H(A,B)2 −Ke(A,B) =1

4

(k1(A,B) − k2(A,B)

)2.

The explicit dependence of the fundamental pair (A,B) in the above notationswill be used only when strictly necessary and it helps to avoid misunderstandings,that is, in general we will just write H, Ke, k1, k2 and q for the curvatures definedabove.

We say that the fundamental pair (A,B) is umbilical at the point p ∈ M , ifq(p) = 0, and that is minimal at p, if H(p) = 0.

Using a local isothermal chart on M , we have that all equalities in Subsection1.1.2 hold for a fundamental pair. Furthermore, the Codazzi tensor vanishes iden-tically if, and only if, (1.10) holds. Thus, we have the next theorem, which can beconsidered as an abstract version of the Hopf theorem. It was established by Milnorin [Mil80].

Theorem 1.33. Let (A,B) be a fundamental pair. Then any of the two following

statements imply the third one.

• (A,B) is a Codazzi pair.

• H is constant.

• The (2, 0)−part of B is holomorphic to the conformal structure given by A.


2Elliptic Special Weingarten

Surfaces of Minimal Type

„Calculating does not equal mathematics. It’s a

subsection of it. In years gone by it was the

limiting factor, but computers now allow you to

make the whole of mathematics more

intellectual.

— Conrad Wolfram

In this chapter we will present some results on the theory of Elliptic Special Wein-garten Surfaces of Minimal Type that will be used in the next chapters, followingmainly [ST95; ST93; BS97; RS94; Ale+10] and references therein.

An immersed surface M in the Euclidean space R3 is called a Special Weingartensurface (SW-surface) if its mean curvature, H, and Gaussian curvature, K, satisfy arelation of the form H = f(H2 −K), where f : [0,∞) → R is a continuous functionof class C1 in (0,+∞); in this case the function f is called the associated functionof the surface M . When the function f satisfies that

4t(f ′(t))2 < 1 for all t ∈ (0,∞), (2.1)

a SW-surface is called a Special Weingarted surface of elliptic type, or an EllipticSpecial Weingarten surface (ESW-surface). We will refer to the condition (2.1) asthe elliptic condition for the function f , and in the case that f satisfies it we willsay that f is an elliptic function. An ESW-surface is also denoted in the literature asf -surface, see [ST93; ST99b; ST01], here we will use also this notation.

Elliptic means that, the family of f -surfaces satisfies the maximum principle inthe sense of the Definition 1.23, see [RS94; BS97]; hence, for ESW-surfaces weare able to apply the geometric comparison principle (Theorem 1.22) and the Alek-sandrov reflection method (Lemma 1.26). This fact gives us an analogy betweenESW-surfaces and constant mean curvature surfaces. Actually, ESW-surfaces are di-vided in two classes depending on whether f(0) 6= 0 or f(0) = 0. Note that inthe first case, any sphere of radius 1

|f(0)| is an ESW-surface, and in the second caseany plane is an ESW-surface. We are interested in the second family of solutions,the family containing all planes of R3. Later, the reader will note that this familybehaves as the family of minimal surfaces.

21

Definition 2.1. Let M be an immersed surface in R3. We say that M is an Elliptic Spe-

cial Weingarten surface of minimal type, ESWMT-surface in short, if M is an f -surface

such that f(0) = 0.

Note that, by (1.4), we can rewrite the Weingarten relation using the skew curva-ture as H = f(q).

2.1 General facts

In this section we will recall some general properties of ESWMT-surfaces. Thesefacts begin to show the similarities between ESWMT and minimal surfaces.

Let f be an elliptic function such that f(0) = 0, we denote by S(f) the family ofall ESWMT-surfaces satisfying H = f(q). As we mentioned above, any plane in R3

is an element of S(f), since f(0) = 0.

The first property to mention is the maximum principle. Rosenberg and Sa Earpin [RS94] proved that any pair of ESW-surfaces, satisfying the same Weingartenrelation, satisfies the maximum principle in the sense of Definition 1.21. Anotherproof of this fact can be found in [BS97]. As a consequence we have the nextproposition.

Proposition 2.2. The family S(f) satisfies the maximum principle.

One of the immediately consequences of the last proposition is the non-existenceof elliptic points in ESWMT-surfaces, as we show in the next proposition.

Proposition 2.3. Let M be an ESWMT-surface in R3. Then the Gaussian curvature of

M is non-positive.

Proof. By contradiction, suppose that there exists a point p ∈ M such that K(p) > 0.Then there exists a neighborhood V of p in M such that all points of V are inthe same side of the tangent plane TpM . Since p ∈ M ∩ TpM we conclude thatM ≥p TpM . Due to the fact that both surfaces TpM and M are in S(f), we canapply the geometric comparison principle locally, around p, to conclude that bothsurfaces agree in some neighborhood of p. This allows us to infer that K(p) = 0,which is a contradiction.

The next proposition, taken from [ST93], sets another property of non-flat ESWMT-surfaces: the set of parabolic points cannot have accumulation points.

Proposition 2.4. Let M be an ESWMT-surface in R3. Then either the zeros of the

Gaussian curvature are isolated, or M is contained in a plane.

22 Chapter 2 Elliptic Special Weingarten Surfaces of Minimal Type

Recall that the convex hull of a set in the Euclidean space can be defined asthe smallest closed convex set containing the original set. It is well known thatany compact minimal surface with boundary immersed in R3 is contained in thethe convex hull of its boundary. Considering that this fact relies on the maximumprinciple, we have the same property for ESWMT-surfaces as we display below.

Proposition 2.5. Let M be a compact ESWMT-surface in R3 with boundary. Then M

is contained inside the convex hull of its boundary.

Proof. This is a direct consequence from the fact that S(f) has the maximum prin-ciple property and that any plane is an element of this family. Note that for anyplane we can find a parallel plane far away from M . Then, moving the plane alongits normal line approaching M , the first contact point of the plane with the surfacemust be at the boundary, since the maximum principle does not allow to have a firstcontact point at the interior of M .

One of the consequences of the last proposition is that there is no compact sur-faces without boundary in the family S(f).

2.2 Rotational examples

As examples of ESWMT-surfaces we have all minimal surfaces, this is for thespecific case when f ≡ 0. In general, Sa Earp and Toubiana proved in [ST95] theexistence and uniqueness of rotational ESWMT-surfaces assuming that the functionf satisfies

lim inft→0+

4t(f ′(t)

)2< 1. (2.2)

The first theorem about the existence of rotationally symmetric examples says:

Theorem 2.6 (Existence of rotational ESWMT-surfaces). Let f be an elliptic function

satisfying f(0) = 0 and (2.2), and let τ > 0 be a real number satisfying

1

τ< lim

t→∞

(t− f

(t2)). (2.3)

Then there exists a unique complete ESWMT-surface, Mτ , satisfying H = f(q) such

that Mτ is of revolution. Moreover, the generatrix curve is the graph of a convex

function of class C3, strictly positive, with τ as a global minimum, and symmetric.

Theorem 2.6 determines pretty well the geometry of rotational examples ofESWMT-surfaces. We have two cases, if the generatrix is or is not bounded. In thefirst case, the ESWMT-surface Mτ is between two parallel planes and, in the secondcase, all coordinates functions of Mτ are proper functions. See Figure 2.1.

2.2 Rotational examples 23

τ

Mτ

τ

Mτ

Figure 2.1: Generatrix (left) and complete ESWMT-surface Mτ (right).

Theorem 2.7 (Sa Earp and Toubiana, [ST95]). Let f be an elliptic function satisfying

f(0) = 0 and (2.2), and let τ > 0 be a real number satisfying (2.3). If f is Lipschitz

at 0, then the ESWMT-surface Mτ , obtained by Theorem 2.6, is not asymptotic to any

plane in R3. Moreover, there exists a catenoid C such that outside of a compact set,

Mτ is in the component of R3 − C containing the revolution axis of C.

Theorem 2.8 (Sa Earp and Toubiana, [ST95]). Let f be a non-negative elliptic func-

tion satisfying f(0) = 0 and (2.2). If f is Lipschitz at 0 and satisfies

limt→∞

(t− f

(t2))

= +∞, (2.4)

then, as τ tends to 0, the generatrix of Mτ tends to a ray orthogonal to the rotation

axis.

The main consequence of the last theorem is that the family of ESWMT-surfacesMτ : τ > 0 behaves quite similar to the family of minimal catenoids. This means,in addition with the geometric comparison principle, that we have a half-spacetheorem for ESWMT-surfaces (with the appropriate requirements for f). The proofof this is the same as for minimal surfaces, see [HM90].

Corollary 2.9 (Half-space theorem for ESWMT-surfaces). Let f be a non-negative

elliptic function Lipschitz at 0 satisfying f(0) = 0, (2.2) and (2.4). Let M be a


complete connected properly immersed ESWMT-surface verifying H = f(q). If M is

contained in a half space of R3, then M is flat.

rotation axis

Figure 2.2: Generatrixes of rotational ESWMT-surfaces when the function f is non-negative, Lipschitz at 0 and satisfies (2.4).

The uniqueness of these examples is obtained in the next theorem.

Theorem 2.10 (Sa Earp and Toubiana, [ST95]). Let f be an elliptic function satisfy-

ing f(0) = 0 and (2.2). If M is a complete ESWMT-surface satisfying H = f(q) and

rotationally symmetric, then there exists τ > 0 such that M is the surface of revolution

Mτ determined in Theorem 2.6.

2.3 Special Codazzi pairs

We start this section recalling that a pair (A,B) of symmetric bilinear forms de-fined on M is called a fundamental pair if A is positive definite, and it is called aCodazzi pair if the Codazzi tensor, Definition 1.30, associated to this pair vanishesidentically.

Consider a SW-surface M satisfying a relation of the form H = f(q) and thefundamental pair (I, II) given by its first and second fundamental forms. Aroundany non umbilical point or any interior umbilical point, there exist local doublyorthogonal parameters (u, v) given by lines of curvature such that we are able torewrite the Codazzi pair as:

I = Edu2 +Gdv2 and II = k1Edu2 + k2Gdv

2,

where k1 and k2 are the principal curvatures of the surface M . Note that withoutloss of generality we can assume that k1 ≥ k2.

In these coordinates, the Codazzi equation satisfied by the fundamental pair isequivalent to the following equations

(k1)vE = −tEv, and (k2)uG = tGu, (2.5)

where t = 12(k1 − k2). Note that t2 = q = H2 −K.

2.3 Special Codazzi pairs 25

Now we will use the relation H = f(H2 −K) in order to get a new Codazzi pairon the SW-surface M . Set

φ(r) =

∫ r

02f ′(s2)ds, (2.6)

and define the following symmetric bilinear forms

If = eφ(t)Edu2 + e−φ(t)Gdv2 and IIf = teφ(t)Edu2 − te−φ(t)Gdv2. (2.7)

The bilinear form If is positive definite since the functions E, G, eφ(t) and e−φ(t) arepositive on M , and clearly the bilinear form IIf is symmetric. Therefore (If , IIf ) isa fundamental pair.

The metric If was defined by Bryant, see [Bry11; Ale+10]. Furthermore, he usedthe complex structure of If to find a holomorphic quadratic form defined globallyon M , whose zeros are the umbilic points of the immersion. This quadratic formallowed Bryant to prove a Hopf type theorem for SW-surfaces.

We are going to use the notation Hf for the mean curvature of the fundamentalpair (If , IIf ), that is, Hf = H(If , IIf ). We will simplify in the same way thenotation for the extrinsic and skew curvature of this new fundamental pair. A quickcomputation shows that these curvatures are

Hf = 0, (2.8)

Kf = −q, (2.9)

qf = q (2.10)

Lemma 2.11. If M is a SW-surface satisfying H = f(H2 − K), then (If , IIf ) is a

Codazzi pair on M .

Proof. The Codazzi tensor associated to the pair (If , IIf ) vanishes identically if, andonly if, the following equalities hold

(teφ(t)E)v = (eφ(t)E)vHf (2.11)

(−te−φ(t)G)u = (e−φ(t)G)uHf . (2.12)

Since Hf = 0, we will show that (teφ(t)E)v = (−te−φ(t)G)u = 0. In the next compu-tation we will use (2.5) and the equalities φ′(t) = 2f ′(t2) and Hv = 2f ′(t2)ttv, thelast one obtained by differentiate the Weingarten relation.

(teφ(t)E)v = eφ(t)(tvE + tφ′(t)tvE + tEv

)

= eφ(t)E(tv + 2f ′(t2)ttv − (k1)v

)

= eφ(t)E(t+H − k1

)

v

= 0.

To prove the other equality, we use an analogous computation.


Remark 2.12. The pair (If , IIf ) can be defined globally in M ; note that in an umbi-

lical point, If = I and IIf ≡ 0. In fact, we can define this pair without using doubly

orthogonal coordinates.

The last lemma allows us to apply Theorem 1.33 to the fundamental pair (If , IIf ),then we conclude that the (2, 0)−part of IIf is holomorphic for the conformal struc-ture given by If . From Lemma 1.2, we already know that the zeros of this holomor-phic form are the umbilic points of the fundamental pair (If , IIf ), this is, the pointswhere qf = 0, but (2.10) shows that they are the umbilic points of the immersion.

Consider an isothermal parameter µ to If around a non umbilical point, then byLemma 1.2 we can write

If = 2λ|dµ|2 and IIf = Qdµ2 + Qdµ2,

since Hf = 0. The fact that Qdµ2 is holomorphic implies that there exists a con-formal parameter z for If such that cQdµ2 = dz2 for a non-zero constant c, see[Mil80]. If we consider c = 2, we get that 2|Q||dµ|2 = |dz|2 is a flat metric.

By Lemma 1.2, |Q|2 = qλ2 and then

λ =|Q|√q

=|Q|t,

which means that tIf = 2|Q||dµ|2.

We summarize this in the next lemma.

Lemma 2.13. Let M be a SW-surface satisfying a relation of the form H = f(q). The

metric√qIf is flat on M \Mu, where Mu is the set of umbilic points of M . Moreover,

there exists a local conformal parameter z such that

√qIf = |dz|2 and IIf =

1

2

(dz2 + dz2

).

2.4 The Gauss map of Special Weingarten surfaces

Let M be an oriented SW-surface satisfying a relation of the form H = f(q). Inthe last section we have defined the Codazzi pair (If , IIf ), away from umbilicalpoints, using a primitive of the function 2f ′(s2

). Actually, the metric If can be

well defined at umbilic points whenever the function 1t sinhφ(t) can be extended to

t = 0. To justify this assertion, we will define the fundamental pair in an equivalentway. Consider the functions

a = cosh φ(t) − H

tsinhφ(t), b = 1

t sinhφ(t) (2.13)

c = t sinhφ(t) −H cosh φ(t), d = cosh φ(t). (2.14)

2.4 The Gauss map of Special Weingarten surfaces 27

A straight computation exhibits that

If = aI + bII and IIf = cI + dII. (2.15)

Note that ad− bc = 1, so we can recover the first and second fundamental forms as

I = dIf − bIIf and II = −cIf + aIIf . (2.16)

Using the conformal parameter obtained in Lemma 2.13, we rewrite (2.16) as

I =1

2

(−b dz2 + 2

d

t|dz|2 − b dz2

),

II =1

2

(adz2 − 2

c

t|dz|2 + adz2

).

With the notation of the Subsection 1.1.2, we can write

P = − b

2, λ =

d

2t, Q =

a

2and ρ =

c

2t. (2.17)

By Lemma 1.7 we know that −Nz = H∂z + t∂z and that this vector field vanishesonly at umbilical points. Since I(Nz, Nz) = −1

2(c − aH), the quantity c − aH

vanishes only when q = 0.

We are going to compute the norm of the vector field ∂z ∧ ∂z.

cos (∡(∂z, ∂z)) =I(∂z , ∂z)

|∂z||∂z | =λ

|P | =d

tb,

then

sin (∡(∂z, ∂z)) =√

1 − cos2 (∡(∂z, ∂z)) =

√t2b2 − d2

tb=

i

sinhφ(t),

finally,

|∂z ∧ ∂z| = |∂z||∂z | sin (∡(∂z, ∂z)) = |P | i

sinh φ(t)=

bi

2 sinh φ(t)=

i

2t.

Since N = 1|∂z∧∂z |∂z ∧ ∂z, we conclude that

I(N, ∂z ∧ ∂z) = |∂z ∧ ∂z| =i

2t. (2.18)

There exist complex functions α and β such that −iN ∧ ∂z = α∂z + β∂z. Henceusing (2.18) we obtain

0 = I(iN ∧ ∂z, ∂z) = I(α∂z + β∂z, ∂z) = Pα+ λβ,

1

2t= −iI(N, ∂z ∧ ∂z) = I(−iN ∧ ∂z, ∂z) = I(α∂z + β∂z, ∂z) = λα+ P β.


Solving the linear system we have that

α =−λ 1

2t

|P |2 − λ2=

d

d2 − t2b2= d and β =

P 12t

|P |2 − λ2=

tb

d2 − t2b2= tb.

Summarizing,

iN ∧ ∂z = − cosh φ(t)∂z − sinhφ(t)∂z , (2.19)

iN ∧ ∂z = sinhφ(t)∂z + cosh φ(t)∂z . (2.20)

The equality (2.20) is obtained from (2.19) by conjugation.

Lemma 2.14. iN ∧Nz = c∂z + ta∂z.

Proof. By Lemma 1.7 and equalities (2.19) and (2.20) we have

iN ∧Nz = i(N ∧ (H∂z + t∂z)

)

= H(iN ∧ ∂z) + t(iN ∧ ∂z)

= −H(cosh φ(t)∂z + sinhφ(t)∂z) + t(sinhφ(t)∂z + cosh φ(t)∂z)

= (t sinhφ(t) −H cosh φ(t))∂z + (t cosh φ(t) −H sinhφ(t))∂z

= c∂z + ta∂z.

Now we have all the elements to recover the natural basis induced by the immer-sion in terms of Nz and iN ∧Nz.

Lemma 2.15. ∂z =1

c− aH

(aNz + iN ∧Nz

).

Proof. Immediate from Lemma 1.7 and Lemma 2.14.

Recalling the expressions for N , Nz and iN ∧ Nz in terms of the Gauss mapof the surface M obtained in the Subsection 1.2.2, we conclude, after a directlycomputation, that

Theorem 2.16. ∂z =a− 1

c− aHgzξ +

a+ 1

c− aHgz ξ.

Following the ideas presented in [GM00], imposing that ∂z(∂z) = ∂z(∂z) weobtain a complete integrability condition from Theorem 2.16. We synthesize this inthe next theorem.

Theorem 2.17. Let M be an oriented SW-surface satisfying a relation of the form

H = f(q), then the Gauss map g satisfies

2

c− aH

(gzz − 2gzgz

g

1 + gg

)= −

(a+ 1

c− aH

)

zgz +

(a− 1

c− aH

)

zgz, (2.21)

where the functions a and c are given by (2.13) and (2.14) respectively.


Proof. In order to simplify the following equations, we will denote x = (x1,x2,x3),A = a−1

c−aH and B = a+1c−aH .

From Theorem 2.16 we get that

(x1)zz = Az1 − g2

(1 + gg)2gz−2A

g + g

(1 + gg)3gzgz − 2A

g(1 − g2)

(1 + gg)3gz gz +A

1 − g2

(1 + gg)2gzz

+Bz1 − g2

(1 + gg)2gz−2B

g + g

(1 + gg)3gz gz − 2B

g(1 − g2)

(1 + gg)3gzgz +B

1 − g2

(1 + gg)2gzz,

(x1)zz = Az1 − g2

(1 + gg)2gz−2A

g + g

(1 + gg)3gzgz − 2A

g(1 − g2)

(1 + gg)3gzgz +A

1 − g2

(1 + gg)2gzz

+Bz1 − g2

(1 + gg)2gz−2B

g + g

(1 + gg)3gz gz − 2B

g(1 − g2)

(1 + gg)3gz gz +B

1 − g2

(1 + gg)2gzz,

(x2)zz = iAz1 + g2

(1 + gg)2gz+2iA

g − g

(1 + gg)3gzgz − 2iA

g + g3

(1 + gg)3gz gz + iA

1 + g2

(1 + gg)2gzz

−iBz1 + g2

(1 + gg)2gz+2iB

g − g

(1 + gg)3gz gz + 2iB

g + g3

(1 + gg)3gzgz − iB

1 + g2

(1 + gg)2gzz,

(x2)zz = −iAz1 + g2

(1 + gg)2gz+2iA

g − g

(1 + gg)3gzgz + 2iA

g + g3

(1 + gg)3gzgz − iA

1 + g2

(1 + gg)2gzz

+iBz1 + g2

(1 + gg)2gz+2iB

g − g

(1 + gg)3gz gz − 2iB

g + g3

(1 + gg)3gz gz + iB

1 + g2

(1 + gg)2gzz,

(x3)zz = 2Azg

(1 + gg)2gz+2A

1 − gg

(1 + gg)3gzgz − 4A

g2

(1 + gg)3gz gz + 2A

g

(1 + gg)2gzz

+2Bzg

(1 + gg)2gz+2B

1 − gg

(1 + gg)3gz gz − 4B

g2

(1 + gg)3gzgz + 2B

g

(1 + gg)2gzz,

(x3)zz = 2Azg

(1 + gg)2gz+2A

1 − gg

(1 + gg)3gzgz − 4A

g2

(1 + gg)3gzgz + 2A

g

(1 + gg)2gzz

+2Bzg

(1 + gg)2gz+2B

1 − gg

(1 + gg)3gz gz − 4B

g2

(1 + gg)3gz gz + 2B

g

(1 + gg)2gzz.

From the equations above, we infer that (xi)zz = (xi)zz (i = 1, 2, 3) if, and onlyif,

0 = (1 − g2)(Az gz −Bz gz

)− (1 − g2)

(Azgz −Bzgz

)

+4

(c− aH)(1 + gg)

((g − g3)gz gz − (g − g3)gzgz

)(2.22)

+2

c− aH

((1 − g2)gzz − (1 − g2)gzz

),


0 = (1 + g2)(Az gz −Bz gz

)+ (1 + g2)

(Azgz −Bzgz

)

+4

(c− aH)(1 + gg)

((g + g3)gz gz + (g + g3)gzgz

)(2.23)

− 2

c− aH

((1 + g2)gzz + (1 + g2)gzz

),

0 = g(Az gz −Bz gz

)− g

(Azgz −Bzgz

)

+4

(c− aH)(1 + gg)

(g2gz gz − g2gzgz

)(2.24)

+2

c− aH

(− ggzz + ggzz

).

If we take (2.22) minus (2.23) plus 2g times (2.24), then we obtain that

0 = −(1 + gg)(Azgz −Bzgz

)− 4

(c− aH)(1 + gg)g(1 + gg)gzgz +

2

c− aH(1 + gg)gzz

Rewriting the above equation we have (2.21).


3Asymptotic behaviour of ends of

Elliptic Special Weingarten

Surfaces of Minimal Type

„Mathematics is not a careful march down a

well-cleared highway, but a journey into a

strange wilderness, where the explorers often get

lost. Rigour should be a signal to the historian

that the maps have been made, and the real

explorers have gone elsewhere.

— William Anglin

This chapter is devoted to study the ends of an ESWMT-surface with finite totalsecond fundamental form. Roughly speaking, we want to prove that those endsare regular at infinity, which means that the growth of each end is at most logarith-mic. This is the situation for minimal surfaces, in fact, being regular at infinity isequivalent to have finite total curvature; this was proved by Schoen in [Sch83].

We begin with the concept of regular at infinity and the equivalence with finitetotal curvature for minimal surfaces.

3.1 Regular at infinity for minimal surfaces

Although Schoen developed this concept for minimal hypersurfaces, see [Sch83],the definition can be used for any hypersurface.

Definition 3.1. A complete immersion f : Mn → Rn+1 is said to be regular at infi-

nity if there is a compact subset K ⊂ M such that M \ K consists of r components

M1, · · · ,Mr such that each Mi is the graph of a function ui with bounded slope over

the exterior of a bounded region in some hyperplane Πi. Moreover, if x1, · · · , xn are

the coordinates in Πi, we require the ui’s have the following asymptotic behaviour for

|x| large and n = 2:

ui(x) = a log |x| + b+c1x1

|x|2 +c2x2

|x|2 +O(|x|−2); (3.1)

while for n ≥ 3 we require

ui(x) = a+ b|x|2−n +n∑

i=1

cjxj |x|−n +O(|x|−n), (3.2)

33

for constants a, b, cj depending on i.

The components Mi in the last definition are the ends of the hypersurface M .Observe that this definition requires embedded ends, but does not prohibit the possi-bility that they intersect.

As we mentioned at the beginning of this chapter, this definition, for minimalsurfaces, is related to finite total curvature. More precisely,

Theorem 3.2 (Schoen, [Sch83]). A complete minimal immersion f : M2 → R3 is

regular at infinity if, and only if, M has finite total curvature and each end of M is

embedded.

The proof presented by Schoen depends strongly on the fact that M is a minimalsurface, especially it depends on the Weierstrass representation. This is the firstlimitation to deal in order to generalize the classification obtained by Schoen toESWMT-surfaces.

For higher dimensions, n ≥ 3, it is proven that regularity at infinity is a con-sequence of minimality and bounded slope. This theorem and its proof have adifferent perspective in contrast to the case of surfaces.

Theorem 3.3 (Schoen, [Sch83]). Assume n ≥ 3, and f : Mn → Rn+1 is a mini-

mal immersion with the property that M \ K, for some compact K, is the union of

M1, · · · ,Mr where each Mi is a graph of bounded slope over the exterior of a bounded

region over a hyperplane Πi. Then M is regular at infinity.

The main idea of the proof is to use the bounded slope to show that the tangentplanes have some “limit plane” at infinity and to obtain a nice growth of the heightof each end. Using elliptic theory, the desired expansion of the functions modelingeach end is achieved.

With this characterization we get,

Theorem 3.4 (Schoen, [Sch83]). The only connected complete minimal immersions

f : Mn → Rn+1, which are regular at infinity and have two ends, are the catenoids.

3.2 ESWMT-surfaces with finite total second

fundamental form

First we are going to prove that the ends of an ESWMT-surface with finite totalsecond fundamental form are, outside of some compact set, graphs of functionsdefined over planes orthogonal to the Gauss map at infinity. We will use resultspresented in Section 1.3.

34 Chapter 3 Asymptotic behaviour of ends of ESWMT-surfaces

Theorem 3.5. Let M be a complete connected oriented ESWMT-surface immersed in

R3. If M has finite total second fundamental form, then M is of finite topology, pro-

perly immersed, the Gauss map extends continuously at infinity and any end is a

multigraph over some plane. Moreover, if an end is embedded, this end is a graph over

some plane.

Proof. By Theorem 1.16, we get that M has finite total curvature, then using Theo-rem 1.14, we conclude that M is of finite topology. Additionally, Proposition 2.3says that K ≤ 0, hence Theorem 1.17 gives that M is properly immersed and theGauss map extends continuously at infinity. Let E be an end of M and supposethat the Gauss map goes (at this end) to the unitary vector ν ∈ R3 at infinity, thenconsidering the plane Πν = x ∈ R3 : 〈x, ν〉 = 0 we infer that the tangent planesare uniformly near to Πν going far out on E. This last information guarantiesthat the projection of E over the plane Πν is a covering map outside a big enoughcylinder, i.e., there exists R > 0 such that the projection pν : E \ Cν(R) → Πν \BΠν

(R) defined by pν(x) = x − 〈x, ν〉ν is a covering map, where Cν(R) = x ∈R3 : ||pν(x)|| ≤ R and BΠν

(R) = x ∈ Πν : ||x|| ≤ R. Since M is properlyimmersed, E cannot be infinitely sheeted, thereby E (outside the compact E ∩Cν(R)) is a multigraph. Besides, if E is embedded, then E \ Cν(R) is a graph overΠν \BΠν

(R).

From now on, we follow the notation of the last proof and we work on an em-bedded end E of M , which is actually a graph over a domain contained in theplane Πν and such that the tangent planes are uniformly near to Πν . We denote byN : E → S2 the Gauss map and θ : E → [−1, 1] ⊂ R the angle function defined byθ(x) = 〈N(x), ν〉 for each x ∈ E. From Theorem 3.5, there exists 0 < ǫ < 1 suchthat 1 − ǫ ≤ θ(x) ≤ 1 for all x ∈ E. Let ι : E → R3 be the inclusion map, thenit is well known that ∆Eι = 2H, where H is the mean curvature vector of E, thatis, H = HN , H the mean curvature and ∆E the Laplacian operator on E. Multi-plying by ν this identity we get ∆E〈ι, ν〉 = 2Hθ. Since E is a graph, we can writeE = x+φ(x)ν : x ∈ Πν \BΠν

(R) for some smooth function φ : Πν \BΠν(R) → R,

and then abusing notation we write 〈ι, ν〉 = φ and ∆Eφ = 2Hθ.

Theorem 1.14 says that the end E is conformally equivalent to the punctured diskD = z ∈ C : 0 < |z| ≤ 1, that is, there exists a diffeomorphism α : D → E suchthat the metrics g0 and g1 = α∗g are pointwise conformal. Here we are denoting asg0 and g, the flat metric in D and the Euclidean metric restricted to E, respectively.It follows that there exists a smooth positive function λ : D → R such that we canwrite g1 = λg0, and, after a straightforward computation, ∆g1 = λ−1∆g0. Definethe function u : D → R by u(z) = 〈α(z), ν〉, due to the fact that α is an isometryfrom (D, g1) onto (E, g) we have that ∆g1u = (∆E〈ι, ν〉) α; joining this with therelation between the Laplacians and the expression obtained for ∆Eφ we achieve

∆g0u = 2λHθ. (3.3)

3.2 ESWMT-surfaces with finite total second fundamental form 35

The right hand side of (3.3) must be read as a function on D, it means that thereis an implicit composition with the diffeomorphism α.

We think on the function u as the height of the end E, over the plane Πν , givenby the conformal immersion α : D → E, see Figure 3.1. In the following sectionwe will study the growth near the origin of this height, and in Section 3.4 we willprove that u(z) is asymptotically radial as |z| → 0.

bb

αz α(z)

ν

D

Πν

E

u(z)

∂E

Figure 3.1: Parametrizing the end E with a conformal parameter.

3.3 Growth of an ESWMT-end

The first claim about the height function u of an end E is that, after a verticaltranslation of the end E, either u ≥ 0 or u ≤ 0. This means that the end E cannotgrow infinitely in both directions ν and −ν simultaneously. This fact is intuitivelytrue, but it is no trivial.

We would like to emphasize that there are harmonic immersions of compactRiemann surfaces with finitely many punctures that are complete, properly em-bedded, negatively curved, with finite total curvature and highly complicated ends,see [Con+15; AL13]. Some of those examples are presented in Figure 3.2. Al-though the family of harmonic surfaces has similarities with the family of minimalsurfaces, for example harmonic surfaces satisfy the maximum principle and havenon-positive Gaussian curvature; some differences appear, for instance there areembedded examples of any genus with just one end.

We will devote this section to prove the claim we made above, Corollary 3.12.The proof presented here is based principally on the work of Chan, see [Cha00;CT01], and Hopf, see [Hop50b].

With the purpose of fixing some notations and hypothesis, let E be an embeddedend of an ESWMT-surface M immersed in R3 with finite total second fundamentalform. We already know that the Gauss map goes, up to a rigid motion, to the unitaryvector ν = (0, 0, 1) at infinity and that, outside some cylinder Cν(R), the end E isthe graph of a function φ : R2 \B(R) → R.


Figure 3.2: Examples of properly embedded harmonic surfaces with finite total curvature1.

Consider a circle γ contained in the domain of φ. Let Pγ denote the linear functionthat best approximates the function φ on the circle γ in the supremum norm. If wedefine

m(γ) = infa,b,c∈R

sup

(x,y)∈γ|φ(x, y) − (ax+ by + c)|

, (3.4)

then m(γ) = sup(x,y)∈γ

|φ(x, y) − Pγ(x, y)|. Note that m(γ) = 0 if, and only if, φ(γ) is a

torsion free curve.

The next lemma, under the hypothesis that φ is not linear, guaranties the exis-tence of a circle such that its best approximating plane is not horizontal.

Lemma 3.6 (Chan and Treibergs, [CT01]). If φ is not a linear function, there exists

a circle γ0 surrounding ∂B(R) such that m(γ0) 6= 0 and ∇Pγ0 6= 0.

The Tchebychev equioscillation theorem, see [Ach92, p. 57], says that if a poly-nomial of degree n is the best approximation of a function defined on a compactinterval, in the supremum norm, then there exist n+2 ordered points on the intervalwhere this distance is attained and the polynomial oscillates around the function.There is a special version of the Tchebychev equioscillation theorem for periodicfunctions approximated by trigonometric functions, see [Ach92, pp. 64–65]. Thisversion immediately guaranties the following lemma.

Lemma 3.7. For any circle γ contained in the domain of φ, there exist at least

four points, ordered as, β+1 , β−

1 , β+2 , β−

2 along γ, such that for i = 1, 2 we have

φ(β+i ) = Pγ(β+

i ) +m(γ) and φ(β−i ) = Pγ(β−

i ) −m(γ).

This lemma means that on the circle γ the function φ − Pγ alternates betweenm(γ) and −m(γ) at the points β±

i .

1From http://www.indiana.edu/~minimal/archive/Harmonic/index.html, by Weber.

3.3 Growth of an ESWMT-end 37

b

b

b

b

Pγ(γ)

φ(γ)

β+1

β−1

β+2

β−2

γ

bb b b

Pγ(γ)

φ(γ)

β+1β−

1 β+2 β−

2

γ

Figure 3.3: The Tchebychev theorem.

The next lemma is due to Bernstein.

Lemma 3.8 (Bernstein, [Ber15]). Let φ be of class C2 in a connected open set Ω and

let φxxφyy − φ2xy ≤ 0 in Ω. Assume φ > 0 in Ω and φ = 0 on ∂Ω. Suppose that Ω can

be placed in a sector of angle less than π. Let M(r) be the maximum of φ on the part

of the circle of radius r, centered at some fixed point, contained in Ω. Then there exist

a positive constant c such that M(r) > cr for all r sufficiently large.

The original proof of Bernstein contained a gap. Hopf in [Hop50a] proved atopological lemma, which is the key to fulfill the gap detected in Bernstein’s work.This lemma is also important in the proof of our result and it reads as follows.

Lemma 3.9 (Hopf, [Hop50a]). If Ω ⊂ Rn is a bounded open set and Ω′ ⊂ Ω is

a connected open subset whose boundary has at least one point in common with ∂Ω,

then the set C of all of this common boundary points contains a set accessible from

within Ω. If C is the union of two disjoint closed sets then each of these sets contains a

set accessible from within Ω.

We are ready to state the main result of this section.

Lemma 3.10 (Growth lemma). Let E be an end of an immersed ESWMT-surface.

Suppose there exists R > 0 such that E is obtained as the graph of a C2 class function

φ defined in R2 \ B(R). Suppose also that |∇φ| → 0 as r → ∞, where r is the

distance from an arbitrary point to the origin. Then at least one of the quantities

inf|(x,y)|≥R

φ(x, y) or sup|(x,y)|≥R

φ(x, y) is finite.

Proof. The Gaussian curvature K of E is then given by

K =φxxφyy − φ2

xy

(1 + φ2x + φ2

y)2,


therefore by Proposition 2.3 the function φ satisfies φxxφyy −φ2xy ≤ 0. If K vanishes

identically, then E is contained in a plane and, hence, φ is linear. From the hypo-thesis on the gradient of φ, we conclude that φ ≡ 0, and the proposition is proven.Consequently we will consider that K does not vanish identically.

Now, assume that

inf|(x,y)|≥R

φ(x, y) = −∞ and sup|(x,y)|≥R

φ(x, y) = ∞, (3.5)

with the purpose of obtaining a contradiction.

Claim A. The function φ has sublinear growth.

Proof. For r ≥ R, let γ(r) be the circle centered at the origin and radiusr, m(r) and M(r) be the minimum and the maximum value of φ on γ(r)

respectively. Taking r big enough, (3.5) implies that

m(r) < min 0,m(R) and M(r) > max 0,M(R) .

By Lemma 3.7, there exist ordered points p1, q1, p2, q2 in γ(r) such thatφ(p1) = m(r), φ(p2) = M(r) and φ(q1) = φ(q2) = 0. By the mean valuetheorem, there exist points w1, w2 in γ(r) such that

−m(r) = |φ(p1) − φ(q1)| ≤ πr|∇φ(w1)|M(r) = |φ(p2) − φ(q2)| ≤ πr|∇φ(w2)|.

Thus, for any p in γ(r), we have

−rǫ(r) ≤ −πr|∇φ(w1)| ≤ m(r) ≤ φ(p) ≤ M(r) ≤ πr|∇φ(w2)| ≤ rǫ(r),

where ǫ(r) =1

πmaxγ(r)

|∇φ|.

To obtain a non-increasing function ǫ(r) and the strict inequality it issufficient to change ǫ(r) by supǫ(r) : r ≥ r + 1

r2 ; this proves theclaim.

From now on, we redefine R, if necessary, in order to argue that the function φ

has sublinear growth in R2 \B(R).

According to Lemma 3.6, there exists a circle γ0 of radius r0 > R looping around∂B(R) such that m(γ0) 6= 0 and |∇Pγ0 | 6= 0. After a translation and a rotation inthe domain of φ, we can assume that the center of γ0 is the origin and Pγ0 = q0y

with q0 > 0. We would like to emphasize that all points of the set φ(γ0) are interiorpoints of the end E. Let B denote the set obtained after the translation and rotationof B(R).


We define the function ψ(x, y) = φ(x, y) − Pγ0(x, y) for any (x, y) ∈ R2 \B. Notethat ψxxψyy −ψ2

xy ≤ 0. From Lemma 3.7 there exist four points, ordered as, β+1 , β−

1 ,β+

2 , β−2 along γ0 such that

ψ(β+1 ) = ψ(β+

2 ) = m(γ0) > 0 and ψ(β−1 ) = ψ(β−

2 ) = −m(γ0) < 0.

Let r > r0 big enough such that ǫ(r) < q0 for all r ≥ r. We define the subset G ofR2 as

G = B(r) ∪

(x, y) ∈ R2 \B(r) : |y| < rǫ(r)

q0

.

Let us analyze the set G in each quadrant of R2. On the first quadrant, using polarcoordinates, the equation |y| = rǫ(r)

q0becomes sin θ = ǫ(r)

q0. Therefore outside B(r),

for each r ≥ r, there is a unique θ satisfying the last equation, moreover, since ǫ(r)q0

is non-increasing, the curve determined by the initial equation is a graph over thex−line (outside B(r)), see Figure 3.4.

b

b

ǫ(r)q0

ǫ(r)q0

1 r r x−line

b

y = rǫ(r)q0

r x−line

Figure 3.4: The set ∂G in the first quadrant.

We must observe that ∂G is symmetric with respect the x-axis and y-axis. There-fore, ∂G is completely determined by the points in the curve solution to |y| = rǫ(r)

q0

contained in the first quadrant, see Figure 3.5.

r

G

L−

L+

Figure 3.5: The set G.

We have obtained that the set G is bounded by two curves L+ and L−. The curveL− is the union of the curves solution of |y| = rǫ(r)

q0in the first and the second

quadrants and the arc of ∂B(r) (in these quadrants) joining these curves. This arc


can be described as the set of points (x, y) ∈ ∂B(r) such that y ≥ rǫ(r)q0

, see the leftside of Figure 3.4 . The curve L+ is the reflection of L− with respect to the x−axis.A sketch of the set G is presented in the Figure 3.5.

On the one hand, any (x, y) ∈ L− satisfies either y = rǫ(r)q0

or y ≥ rǫ(r)q0

. Thereforeon L− either ψ(x, y) = φ(x, y) − q0y = φ(x, y) − rǫ(r) < 0 or |(x, y)| = r andψ(x, y) = φ(x, y) − q0y ≤ φ(x, y) − rǫ(r) < 0. Thus, the function ψ is negativeon L−. Analogously we can prove that ψ is positive on L+. On the other hand,any point (x, y) above the curve L− satisfies y > rǫ(r)

q0, then we can conclude that

ψ(x, y) = φ(x, y) − q0y < rǫ(r) − rǫ(r) = 0. In the same way, any point (x, y) belowthe curve L+ satisfies ψ(x, y) > 0.

This shows that the set (x, y) ∈ R2 \B : ψ(x, y) = 0 must be strictly containedin G. Clearly this set is not empty since ψ alternates between −m(γ0) and m(γ0) on∂B(r0). Let us denote the sets

Ω+ = (x, y) ∈ R2 \B(r0) : ψ(x, y) > m(γ0),

Ω− = (x, y) ∈ R2 \B(r0) : ψ(x, y) < −m(γ0).

Claim B. Neither Ω+ nor Ω− is empty.

Proof. Suppose that the set Ω+ is empty; then ψ(x, y) ≤ m(γ0) for all(x, y) ∈ R2 \ B(r0). Recalling that ψ(x, y) = φ(x, y) − q0y we infer thatφ(x, y) ≤ q0y + m(γ0), this means that the end E is placed below theplane q0y+m(γ0) which is above plane parallel to Pγ0 at distance m(γ0),see Figure 3.6. The intersection between the end and the plane has nointerior points of the end, by the geometric comparison principle. Con-sequently, the intersection occurs only on ∂B(r) and this intersection istransversal, again by the geometric comparison principle. Moreover, asr → ∞, the distance between the end E and the plane q0y+m(γ0) goesto infinity since the function φ has sublinear growth and the Gauss mapgoes to ν at infinity. Now we can decrease the inclination of the planekeeping the end E below until we obtain either an interior first contactpoint between the end and some plane, or a horizontal plane such thatE is below of this plane, see Figure 3.7. The first case cannot happen,by the geometric comparison principle, and the second case contradictsthe assumption (3.5). Analogously, the statement Ω− = ∅ contradictsthe supposition of that E grows also infinitely in the direction −ν.

Claim C. Each component of Ω+ and Ω− is unbounded.

Proof. Suppose that a component of Ω+ is bounded; geometrically thismeans that the part of the end E, corresponding to that component,above the plane q0y + m(γ0) is compact. Hence for a positive constant


c the plane q0y + m(γ0) + c and the end E have an interior contactpoint, again the geometric comparison principle leads to a contradiction.Analogously for the components of Ω−.

γ0

B(r0)

E

Pγ0= q0y

Pγ0+ m(γ0)

Figure 3.6: Assuming that Ω+ is empty.

Figure 3.7: Decreasing the inclination of the plane.

Claim D. Each set Ω+ and Ω− has at least two connected unbounded

components.

Proof. Each component of the sets Ω+ and Ω− is unbounded by ClaimC. Note that β+

1 , β+2 ∈ Ω+ and β−

1 , β−2 ∈ Ω−. Let us denote by Ω±

i theconnected component of Ω± such that β±

i ∈ ∂Ω±i . Assume Ω+

1 = Ω+2 ,

then we can connect the points β+1 and β+

2 by a curve contained inΩ+

1 = Ω+2 , moreover, we can connect these points also by a line segment

contained in B(r0); these curves enclose a compact set containing eitherβ−

1 or β−2 , hence one of the sets Ω−

i is contained in this compact set,thus Ω−

i is empty, since all components are unbounded, see Figure 3.8.Interchanging the roles of Ω+

i and Ω−i , we obtain a similar conclusion.

Then, it is enough to prove that all the sets Ω±i are not empty. Suppose

that Ω+1 = ∅. In this case, the end E in a neighborhood of φ(β+

1 ) isbelow the plane q0y+m(γ0). This situation is avoided by the geometriccomparison principle. The same argument is valid for the other sets.


b b

b

b

γ0

β+1

β−

1

β+2

β−

2

Ω+1

Ω+2

b b

b

b

γ0

β+1

β−

1

β+2

β−

2

Figure 3.8: Assuming that Ω+

1 = Ω+

2 .

Now we will consider the components of the sets

(x, y) ∈ R

2 \B(r0) : ψ(x, y) > 0

and

(x, y) ∈ R2 \B(r0) : ψ(x, y) < 0

.

Each of these sets have at least two components containing the sets Ω±i . We will

denote these components as Ω′±i , then

β+i ∈ Ω+

i ⊂ Ω′+i and β−

i ∈ Ω−i ⊂ Ω′−

i .

Claim E. Each set Ω′±i contains a Jordan curve J±

i (t) = (x±i (t), y±

i (t))

contained in the set G \ B(r0) such that x+i (t) → ∞ as t → ∞ and

x−i (t) → −∞ as t → ∞.

Proof. If any of the sets Ω′±i contains a point of L±, then by the connec-

tedness it must contain the whole curve L±. In this case the curve L±

satisfies the condition required.

Now suppose that some set Ω′±i does not touch the curve L±, this im-

plies that Ω′±i ⊂ G \B(r0). We now add the points +∞ and −∞ to the

boundary of G and consider again the set Ω±i , then clearly

∂G ∩ ∂Ω±i ⊂ −∞,+∞. Assume that +∞ is not in the common

boundary, thereby there exists a big enough positive constant C suchthat Ω±

i ⊂ (x, y) ∈ R2 : x ≤ C

. In this case we can easily find an an-

gle less than π where the set Ω±i can be placed, since ǫ(r)

q0→ 0 as r → ∞.

The Figure 3.9 shows how to build the angle sector.

Applying Lemma 3.8 to the function ψ−m(γ0) on the set Ω±i , we obtain

that for r big enough and some (x, y) with |(x, y)| = r we have

ψ(x, y) −m(γ0) > cr.

However, φ has sublinear growth on Ω±i , i.e.,


ψ(x, y) −m(γ0) = φ(x, y) − q0y −m(γ0)

≤ rǫ(r) + q0|y| −m(γ0)

≤ 2rǫ(r) −m(γ0),

but the above inequalities imply that ǫ(r) > c2 , which is a contradiction

since ǫ(r) → 0 as r → ∞.

b

γ0

C

L−

L+

β±

iΩ±

i

Figure 3.9: The set Ω±

iplaced in an angle sector.

Summarizing, we have proven that ∂G ∩ ∂Ω±i = −∞,+∞ for all sets

Ω±i ; therefore Lemma 3.9 guaranties the existence of the Jordan curves

J±i .

To finish the proof, consider for i = 1, 2 a curve in G that joins +∞ and β+i , also

consider the segment inside B(r0) joining β+1 and β+

2 . The closed curve formed bythese three curves must enclose one of the points β−

i , and, in consequence, theymust enclose one of the sets Ω−

i . This is a contradiction because now −∞ is notaccessible from within Ω−

i , see Figure 3.10. This concludes the proof of the lemma.

bb

b

b

γ0

β+1

β−

1

β+2

β−

2J+

1

J+2G \ B(r0)

+∞−∞

Figure 3.10: The point β−

2 cannot be connected to −∞ by the curve J−

2 .


Theorem 3.11. Let M be a complete connected ESWMT-surface in R3 with finite total

second fundamental and one embedded end. Suppose that the function f associated to

M is non-negative, Lipschitz at 0 and satisfies (2.2). Then M is a plane.

Proof. By Theorem 3.5 and the growth lemma, M lies in a half-space of R3. SinceM has finite total second fundamental form, the skew curvature is bounded on M .Let qM be the maximum value of this curvature and f : [0,∞) → R be the functiondefined as

f(t) =

f(t), if 0 ≤ t ≤ qM

a√t+ b, if t > qM

,

where a = 2√qMf

′(qM ) and b = f(qM) − 2qMf ′(qM ). On the one hand,

f(qM ) = f(qM) and f ′(qM ) = f ′(qM ),

and on the other hand, if t ≥ qM , then

4t(f ′(t)

)2= 4qM

(f ′(qM )

)2< 1.

Summarizing, f is continuous in [0,∞), of class C1 in (0,∞), non-negative,Lipschitz at 0 and satisfies (2.1). Moreover,

limt→∞

((t − f

(t2))

= limt→∞

((1 − a)t − b) = +∞,

because 0 ≤ a < 1. ClearlyM is a f -surface and, by Corollary 2.9, M is a plane.

Corollary 3.12. After a vertical translation of the end E, the function u in Section 3.2

is either non-negative or non-positive.

Proof. If we write α = (x, y, u) for the function α : D → E ⊂ R3 described inthe last section, then we see that u(z) = φ(x(z), y(z)). It follows from the growthlemma, Lemma 3.10, that either

inf0<|z|≤1

u(z) > −∞ or sup0<|z|≤1

u(z) < ∞,

therefore after a vertical translation of the end E, either u ≤ 0 or u ≥ 0 which isthe desired conclusion.

3.4 The height near infinity

Summarizing, let M be a connected oriented ESWMT-surface with finite totalsecond fundamental form, thereby M is of finite topology. Let E be one of itsends, E is conformally a punctured disk, that is, there exists a diffeomorphismα : D → E such that the parameter z ∈ D is conformal. Also, the Gauss mapN extends continuously to the origin, where we assume, after a rigid motion in

3.4 The height near infinity 45

R3, that N(z) tends to ν = (0, 0, 1) as |z| → 0. The function u : D → R definedby u(z) = 〈α(z), ν〉 satisfies the Poisson equation (3.3), which can be written as∆u = 2λθf(q) provided the Weingarten relation H = f(q).

Henceforth, we will consider that u ≥ 0, hence u and ∆u have the same sign, byCorollary 3.12. Note that the case of u ≤ 0 is completely analogous since we canredefine ν as −ν, this will change the sign of the angle function θ and we will havethe same conclusion.

From now on, we will consider f non-negative and Lipschitz. The Lipschitz condi-tion will give that the Laplacian of the height function is integrable over the wholeunit disc, this fact together with the non-negativity of the function will allow us toinfer a first estimate on the growth of the end E near infinity, this is a first charac-terization of the function u around the origin (Lemma 3.15). Let us formalize theseideas.

We now turn to the Lipschitz hypothesis. Note that when |z| goes to 0, the meanand Gaussian curvatures goes to zero, hence the skew curvature goes to zero aswell. In conclusion, in a neighborhood of 0 we obtain that

∆u = 2λθf(q) ≤ 4δλθq = δλθ(|II|2 − 2K), (3.6)

for some positive constant δ.

Recalling that θ ≤ 1 and integrating around the origin, we conclude, for some0 < R < 1 small enough, that

∫

D(R)∆u |dz|2 ≤ δ

∫

D(R)λ(|II|2 − 2K) |dz|2 ≤ δ

∫

E(|II|2 − 2K) dA < ∞, (3.7)

Note that we have used the hypothesis of finite total second fundamental form andTheorem 1.16.

3.4.1 Isolated singularities of subharmonic functions

In this subsection we are going to work with the equation ∆u = ϕ, where u andϕ are non-negative functions defined in D and ϕ is integrable over the whole diskD. Let us denote the sets D(r) = z ∈ C : |z| ≤ r and D(r) = D(r) \ 0, wherer is a positive real number. In the case r = 1 we just write D and D respectivelyas in the previous sections. For any 0 < r1 ≤ r2 we denote by A(r1, r2) the setz ∈ C : r1 ≤ |z| ≤ r2. The ideas and proofs presented here are based principallyon some articles written by Taliaferro, specifically the references [Tal99; Tal06].

For 0 < r ≤ 1 we define the average of u on ∂D(r) as

u(r) =1

2πr

∫

∂D(r)u(x, y) ds.


Note that after a suitable change of coordinates we can rewrite this function as

u(r) =1

2π

∫

∂Du(rx, ry) ds.

Lemma 3.13. If ∆u = ϕ, then for any 0 < r ≤ r1 ≤ 1,

ru′(r) = r1u′(r1) − 1

2π

∫

A(r,r1)ϕ(z) |dz|2. (3.8)

Proof. We have

u′(r) =1

2π

∫

∂D

(∂u

∂x(rx, ry)x+

∂u

∂y(rx, ry)y

)ds

=1

2π

∫

∂D(∇u(rx, ry) · (x, y)) ds

=1

2π

∫

∂D(r)

(∇u(x, y) ·

(x

r,y

r

))ds

=1

2πr

∫

∂D(r)

∂u

∂ηds,

where ∂u∂η is the normal derivative. This implies, for any 0 < r ≤ r1 ≤ 1, that

r1u′(r1) − ru′(r) =

1

2π

(∫

∂D(r1)

∂u

∂ηds−

∫

∂D(r)

∂u

∂ηds

)

=1

2π

∫

∂A(r,r1)

∂u

∂ηds

=1

2π

∫

A(r,r1)∆u(z) |dz|2.

From the hypothesis on the function ϕ, we have

limr→0+

∫

A(r,r1)ϕ(z) |dz|2 ≤

∫

D

ϕ(z) |dz|2 < ∞,

therefore the right-hand side in equation (3.8) tends to some real value β as r goesto zero (which clearly does not depend of r1). This observation allows us to write

ru′(r) = β + o(1), as r → 0+.

Thus u′(r) = βr−1+o(r−1) as r → 0+, this implies by integration that u(r)log r = β+o(1)

as r → 0+. Since limr→0+

r log r = 0, then

ru(r) = βr log r + o(1) as r → 0+,

and we infer that∫

D(ε)u(z) |dz|2 =

∫ ε

02πru(r) dr = O

(ε2 log ε

), as ε → 0+. (3.9)


Consider the function uN : D → R defined by

uN (z) =1

2π

∫

D

log1

|z − ξ|ϕ(ξ) |dξ|2; (3.10)

this function is called the Newtonian potential of ϕ in D. It is well known that∆uN = ϕ, nonetheless, a complete proof of this fact can be found in [GT01; Eva98].Hence, repeating exactly the same steps as above, we conclude that

∫

D(ε)uN (z) |dz|2 = O

(ε2 log ε

), as ε → 0+. (3.11)

Note that the function u− uN is a harmonic function on D.

The next lemma shows the nature of the singularity of a harmonic function, im-posing some condition on the integral of the absolute value of the function.

Lemma 3.14 (Taliaferro, [Tal99]). Suppose v is harmonic in D and that

∫

D(ε)|v(z)| |dz|2 = o (ε) , as ε → 0+.

Then there exists a constant β such that v(z) + β log |z| has a harmonic extension to

D.

We would like to apply the last lemma to the function u− uN since, by (3.9) and(3.11), we have

∫

D(ε)|u(z) − uN (z)| |dz|2 ≤

∫

D(ε)|u(z)| |dz|2 +

∫

D(ε)|uN (z)| |dz|2

=

∫

D(ε)u(z) |dz|2 +

∫

D(ε)uN (z) |dz|2

= O(ε2 log ε

).

In conclusion, we have

Lemma 3.15 (Taliaferro, [Tal99]). Let u and ϕ be non-negative functions defined on

D, such that ∆u = ϕ in D and ϕ is integrable over the whole disk D. Then there

exists a constant β and some continuous function uh : D → R which is harmonic in D,

such that for any z ∈ D we have

u(z) = β log |z| + uh(z) + uN (z), (3.12)

where the function uN is defined by (3.10).

We finalize this section with the following lemma that shows the behaviour of theNewtonian potential of ϕ near the singularity.


Lemma 3.16 (Taliaferro, [Tal06]). Let u and ϕ as in Lemma 3.15. If there exists a

non-negative constant C and 0 < r ≤ 1 such that ϕ(z) ≤ (2|z|)−C for z ∈ D(r), then

the Newtonian potential of ϕ satisfies

uN (z) = o

(log

1

|z|

)as |z| → 0+. (3.13)

3.4.2 Asymptotic behaviour of the height

Here we are going to apply the results of the last subsection to the height functionu = 〈α, ν〉. From Corollary 3.12, (3.3), (3.6) and (3.7) we have guarantied thehypothesis of Lemma 3.15. Then u can be written in the form (3.12).

As we have pointed out above, the mean curvature H goes to 0 as |z| → 0+. Thengiven a non-negative constant C and a positive constant A, for |z| small enough, wecan assume that H(z) ≤ 2−C+1A−1. Therefore, by recalling that the angle functionis bounded above by 1, from (3.3) we get

∆u ≤ 1

2CAλ, (3.14)

where λ is the conformal factor of the metric.

Now, in order to estimate λ, we have the following theorem that is an easy con-sequence of several theorems proven by Finn, see [Fin65; Fin64].

Theorem 3.17 (Finn, [Fin65]). Let M be a complete Riemannian surface with finite

total curvature in a neighborhood E of one of its ideal boundary components. Suppose

that the region of positive curvature has compact support on the surface. Then E

can be mapped conformally onto the punctured disk D in the complex plane. Let

denote by λ|dz|2 the conformal metric. Then there exist constants A and Φ0 such that

λ(z) ≤ A|z|−2Φ0 as |z| → 0+. Moreover Φ0 ≥ −1.

Therefore, using last theorem and (3.14), we conclude that for |z| small enough∆u ≤ (2|z|)−C , where C = max0, 2Φ0. This allows us to use Lemma 3.16 in ourgeometric context and infer that

lim|z|→0+

u(z)

log |z| = β. (3.15)

3.4.3 The Beltrami equation and quasiconformal maps

The Gauss map g of an immersed surface in R3 satisfies a very rich partial diffe-rential equation, the Beltrami equation. In general this equation can be written as

gz = µgz. (3.16)


Although there are a variety of equivalent definitions for a quasiconformal map,here we will consider the definition based on the Beltrami equation. This definitionis taken from [Gut+12, p. 47].

Definition 3.18. Let D ⊂ C be a domain. The function g : D → C is called a qua-

siconformal map if it is a homomeomorphic solution of the Beltrami equation (3.16)with ||µ||∞ < 1.

Given a quasiconformal map g, at points z ∈ D where g is differentiable we definethe value

Kµ(z) =1 + |µ(z)|1 − |µ(z)| = lim

r→0

max|w−z|=r

|g(z) − g(w)|

min|w−z|=r

|g(z) − g(w)| .

This value is called the dilatation of g at z. The maximal dilatation of g is definedas Kg = sup

z∈DKµ(z) and it satisfies

Kg =1 + ||µ||∞1 − ||µ||∞

. (3.17)

It is very common to find the notation Kg-quasiconformal to indicate a quasiconfor-mal map g with maximal dilatation Kg or parameter of quasiconformality Kg, seefor example [Ahl66].

Note that 1-quasiconformal maps are exactly the conformal maps because Kg = 1

only when µ = 0 which means, from the Beltrami equation, that gz = 0, i.e., g isholomorphic. This is precisely what happens when the function g is the Gauss mapof a minimal immersion. In this case we say that g is antiholomorphic since weusually construct g by sterographic projection from (0, 0, 1).

A posthumous paper by Mori [Mor56] studies quasiconformal maps from the unitball into itself such that the origin is fixed. The main result of this paper is knownnowadays as the Mori theorem and it establishes a Hölder condition for the functiong.

Theorem 3.19 (Mori theorem). Let g : D → D be a Kg-quasiconformal map with

g(0) = 0. Then

|g(z) − g(w)| ≤ 16|z − w|1/Kg , (3.18)

for all z,w ∈ D. Furthermore, the constant 16 in (3.18) cannot be replaced by any

smaller constant independent of Kg.

Let us go back to our geometric context, where g is the Gauss map of an ESWMT-surface with finite total second fundamental form. According to Theorem 1.10, theGauss map satisfies the Beltrami equation with µ = Hλ

Q . Using the Weingartenrelation H = f(q) and Lemma 1.2 we conclude that

|µ| =f(q)√q. (3.19)


We can assert that

0 ≤ lim supt→0+

f(t)√t

≤ lim supt→0+

f ′(t)1

2√

t

= lim supt→0+

2√tf ′(t)

=

√lim sup

t→0+

4t(f ′(t)

)2.

Assuming that f is Lipschitz, we conclude that f ′(t) is uniformly bounded, thus

lim supt→0+

4t(f ′(t)

)2= 0.

And therefore,

lim supr→0+

sup|µ(z)| : 0 < |z| < r = lim supt→0+

f(t)√t

= 0.

This means that, redefining the end E if necessary, we can get ||µ||∞ as small as wewant. In particular, ||µ||∞ ≤ σ < 1 for some non-negative real constant σ.

Summarizing, we have proved that if the elliptic function f is Lipschitz, then theGauss map is a Kg-quasiconformal map, where the parameter of quasiconformalitysatisfies

Kg ≤ 1 + σ

1 − σ.

Moreover, we can redefine the end E in order to make the parameter of quasicon-formality be as close to 1 as we wish.

In order to use the Mori theorem, we are going to distinguish for a momentbetween the Gauss map obtained by the stereographic projection from (0, 0,−1)

and (0, 0, 1) writing g for the first case (as it was being) and for the second one.Then we have that (0) = 0 and, by the Mori theorem, satisfies for all z ∈ D

that|(z)| ≤ 16|z|

1−σ

1+σ .

This means that (z) = O (|z|ǫ) as |z| → 0+, where 1 ≥ ǫ = 1−σ1+σ > 0. Since g ¯ = 1

we deduce thatg(z) = O

(|z|−ǫ) as |z| → 0+. (3.20)

Remark 3.20. In order to guaranty (3.20) we just need that the function f satisfies

that

lim supt→0+

4t(f ′(t)

)2< 1. (3.21)

The Lipschitz condition for the function f , as we showed, is stronger than (3.21); we

can also verify this through the function f(t) = 12t sin2

(1√t

). On the other hand, the

Lipschitz condition at zero for f , i.e., there exist a constant C > 0 such that f(t) ≤ Ct


for any t ≥ 0, is not enough to reach (3.21) since we can construct an elliptic function

Lipschitz at zero such that lim supt→0+

4t(f ′(t)

)2= 1.

3.4.4 Regularity at infinity for ESWMT-surfaces

Let E be an embedded end of an ESWMT-surface with finite total second funda-mental form. According to Theorem 3.5, E can be described as a graph of a realvalued function φ defined on the complement of a compact set in a plane, which isorthogonal to the limit of the Gauss map at infinity. We would like to understand theasymptotic behaviour of this function as in the case of minimal surfaces, we expectto find an expression for φ analogous to (3.1). Until now, we have described theend E by a conformal parametrization α : D → E such that the function u = 〈α, ν〉is determined by (3.12) and satisfies u(z) = O (log |z|) as |z| → 0+.

In Lemma 3.15 we have shown that the function uh is harmonic on D.Using an homogeneous expansion for the function uh at 0 ∈ D, see [Axl+01,p. 100], and Lemma 3.16, we can rewrite (3.12) for any z = x+ iy ∈ D as

u(z) = β log |z| + a0 + a1x+ a2y +O(|z|2

)+ o

(log

1

|z|

)as |z| → 0+, (3.22)

for some real constants β, a0, a1, a2. Here, we have used that there are only twolinearly independent homogeneous harmonic polynomials of degree m in two di-mensions, namely, the real and the imaginary part of zm.

In order to establish an easy notation, we will assume that the Gauss map takesthe value ν = (0, 0, 1) at infinity and the function φ is defined on the complementof a compact subset of R2 = (x1, x2, 0) : x1, x2 ∈ R ⊂ R3. Note that with thisconvention the function u is the third coordinate function of α; naming x1 andx2 the other coordinates of the diffeomorphism α we can write α = (x1, x2, u).From this, it is clear that the functions u and φ are essentially the same, in factφ(x1, x2) = u(z(x1, x2)). In order to know how the parameter z depends of x1

and x2 we will use the Kenmotsu representation, Theorem 1.12, to recover thecoordinate functions x1 and x2 in terms of z.

From Theorem 1.9 we have the following equations

∂zx1 = − gz(1 + g2)

H(1 + gg)2,

∂zx2 = − igz(1 − g2)

H(1 + gg)2,

∂zu = − 2gzg

H(1 + gg)2.

Hence we conclude that ∂z(x1 + ix2) = −g∂zu and we can recover x1 + ix2 byintegration from the Kenmotsu representation.


Since u(z) = O(log |z|) as |z| → 0+ by (3.15), we have that ∂zu(z) = O(|z|−1

)as

|z| → 0+. On the other hand, if the elliptic function f is Lipschitz in [0,∞), thenby (3.20), g(z) = O (|z|−ǫ) as |z| → 0+, for some ǫ > 0. We reach upon integrationthat x1 + ix2 = O (|z|−ǫ) as |z| → 0+. Defining r = |x1 + ix2| =

√x2

1 + x22, we

have that |z| = O(r−1/ǫ

)as r → ∞. Again, rewriting the function u, from the

equation (3.22), we achieve the following description of the asymptotic behaviourof the function φ,

φ(x1, x2) = β log r + a0 +a1x1

r2/ǫ+a2x2

r2/ǫ+O

(r−2/ǫ

)+ o (log r) as r → ∞,

for some real constants ǫ, β, a0, a1, a2 (not necessarily the same constants of (3.22))where ǫ > 0.

Summarizing all the results achieved in this chapter, we get

Theorem 3.21. Let M be a complete ESWMT-surface in R3 with finite total second

fundamental form and embedded ends. Suppose that the function f associated to M

is non-negative and Lipschitz. Then each end Ei is the graph of a function φi over the

exterior of a bounded region in some plane Πi. Moreover, if x1, x2 are the coordinates

in Πi, then the function φi have the following asymptotic behaviour for r =√x2

1 + x22

large

φi(x1, x2) = β log r + a0 +a1x1

r2/ǫ+a2x2

r2/ǫ+O

(r−2/ǫ

)+ o (log r) , (3.23)

where ǫ, β, a0, a1, a2 are real constants depending on i and 0 < ǫ ≤ 1.

E

∂E

Figure 3.11: Representation of an end E of an ESWMT-surface satisfying the hypothesis ofTheorem 3.21.

We would like to finish this section by noting that the equality (3.23) can be seenas a generalization of the concept of regular at infinity given by Schoen in [Sch83],Definition 3.1.


4Characterization of

ESWMT-surfaces with two ends

„Mathematics is the greatest adventure of

thought. In other activities we also think,

obviously, but we also have the guidance and

control of empirical observation. In pure

mathematics we navigate through a sea of

abstract ideas, with no compass other than

logic.

— Jesús Mosterín

Los lógicos

In this chapter we prove that a connected, oriented ESWMT-surface M embeddedin R3 with two ends, non-negative mean curvature and finite total second funda-mental form is rotationally symmetric with respect to a fixed line, which means thatthe surface M must be one of the rotational examples Mτ (Theorem 2.7) obtainedby Sa Earp and Toubiana in [ST95]. This result will be a consequence of the asymp-totic behaviour of the ends obtained in Chapter 3 and the Aleksandrov reflectionmethod presented in Subsection 1.4.3.

4.1 Embedded ESWMT-surfaces with two ends

Let M be a complete ESWMT-surface in R3 with finite total second fundamentalform and embedded ends. Then, for each end Ei there exists a unit vector νi suchthat Ei is the graph of a function φi defined over the exterior of a compact set inthe plane Πνi

= x ∈ R3 : 〈x, νi〉 = 0. If we also assume that f is non-negativeand Lipschitz, then each function φi has logarithmic growth by Theorem 3.21. Thisimplies that, if the planes Πνi

are not parallel, then the ends must intersected, whichmeans that the surface M is not embedded.

In particular, when the surface M is embedded and has two ends, we have thatthese ends can be described as graphs of functions defined over the same plane Π.Moreover, the surface M must grow infinitely in both sides of the plane Π, since itcannot be contained in a half-space by the half-space theorem, Corollary 2.9.

Recall that the vectors νi are the continuous extension of the Gauss map to theideal boundary pi, Theorem 1.17. When the vectors νi are parallel, we will saythat the corresponding ends Ei are parallel.

We summarize these facts in the next lemma.

55

Lemma 4.1. Let M be a complete ESWMT-surface embedded in R3 with finite total

second fundamental form and two ends. Suppose that the function f associated to M

is non-negative and Lipschitz, then the ends of M are parallel and the surface must

grow infinitely in opposite directions.

4.2 The Aleksandrov function and tilted planes

In this section we are going to prove that, given an end E of an ESWMT-surfaceas a graph of a function of the form (3.23) and a vertical plane P, the maximumvalue of the Aleksandrov function Λ1 of E respect to the plane P is achieved at∂E.

Without loss of generality we can assume that ∂E ⊂ Πν since we can translatevertically the end E and also redefining the end cutting off the compact part thatis at one side of the plane Πν after the translation. Besides, we can also considerthat the function φ is non-negative, after a reflection through the plane Πν , if nece-ssary.

In order to define the Aleksandrov function, recall Subsection 1.4.3. Let W bethe connected component of the upper half-space of Πν that contains the boundeddomain limited by E ∩ Πν . Consider a plane P parallel to the vector ν = (0, 0, 1)

and the height function respect to the plane Πν defined by h(p) = 〈p, ν〉. We definethe associated Aleksandrov function Λ : [0,∞) → −∞ ∪ R on E as

Λ(ρ) = max Λ1(p) : p ∈ D, h(p) = ρ, (4.1)

where D is the domain of the Aleksandrov function Λ1, (1.24), associated to E.

Note that the assumptions made above give us that, for any ρ ≥ 0, the level setEρ = q ∈ E : h(q) = ρ is a non-empty compact set. Then the function Λ is finitevalued, since the set p ∈ D : h(p) = ρ is non-empty and compact. This situationchanges severely if we tilt the plane P slightly. We will study the consequences, onthe associated Aleksandrov function, caused by the tilting.

Let n be the normal vector of the plane P and consider the vectors νε = ν + εn

and nε = n − εν for a small ε > 0. Note that 〈νε, nε〉 = 0. We will define the tiltedplanes Πνε = p ∈ R3 : 〈p, νε〉 = 0 and Pε being the plane passing through theorigin with normal vector nε. The height function from the plane Πνε is hε definedby hε(p) = 〈p, νε〉. Clearly Pε → P, Πνε → Πν , and hε → h as ε → 0.

Since the end E has logarithmic growth, we will obtain two important facts whenε > 0 is small. The first one is that the height function hε restricted to Dε must attaina minimum value hε

0. The second one is that the Aleksandrov function Λε1 is valued

as −∞ outside a non-empty compact set of Dε. More precisely,

56 Chapter 4 Characterization of ESWMT-surfaces with two ends

Lemma 4.2. Let ε > 0. There exists a non-empty compact set B ⊂ Pε such that for

any p ∈ Dε∩(Pε \B) there exists just first contact point of the set p+tnε : t ∈ R∩Was t decreases from +∞, i.e., Λε

1(p) = −∞.

Proof. We are going to prove this lemma in the case that n = (1, 0, 0). This issufficient, since the expansion for the function φ in Theorem 3.21 is invariant underrotation of the (x1, x2)-coordinates. Let us consider a new coordinate system for R3

defined by(y1, y2, y3) = (x1 − εx3, x2, x3 + εx1). (4.2)

Using these coordinates we get the planes Pε = (y1, y2, y3) ∈ R3 : y1 = 0 andΠνε = (y1, y2, y3) ∈ R3 : y3 = 0. The intersection of the end E with the planeparallel to Πνε at height τ > 0 is given by the curve

Γτ =

(y1, y2, τ) ∈ R

3 : φ

(y1

1 + ε2+

ετ

1 + ε2, y2

)=

τ

1 + ε2− εy1

1 + ε2

.

Let Φ be the function defined as

Φ(y1, y2) = φ

(y1

1 + ε2+

ετ

1 + ε2, y2

)− τ

1 + ε2+

εy1

1 + ε2.

Hence,∂Φ

∂y1(y1, y2) =

1

1 + ε2

∂φ

∂x1(x1, x2) +

ε

1 + ε2, (4.3)

where (x1, x2) =(

y1

1+ε2 + ετ1+ε2 , y2

). Since |∇φ| → 0 as r → ∞, there exists R > 0,

big enough, such that ∂φ∂x1

> −ε for all r > R. On the one hand, let ER be the set ofpoints in E such that r ≤ R, i.e.,

ER = (x1, x2, x3) ∈ E : r =√x2

1 + x22 ≤ R.

Note that ER is a non-empty compact set, since it is the graph of a continuousfunction defined over a compact set. Let ρ be the maximum value of the functionhε on the set ER, then for τ > ρ we have that all points of Γτ satisfy r > R. On theother hand, without any assumption on τ , if (y1, y2, τ) ∈ Γτ satisfies |y2| > R, thenr > R. Summarizing, by (4.3), if (y1, y2, τ) ∈ Γτ is a point satisfying either τ > ρ or|y2| > R, then ∂Φ

∂y1(y1, y2) > 0. This means that if τ > ρ, the entire curve Γτ , in the

(y1, y2)-plane is the graph of a function defined in the variable y2 and, consequently,(in the (y1, y2, y3)-coordinates) for all y2 ∈ R, the line (t, y2, τ) ∈ R3 : t ∈ Rintersects the end E exactly once. We have the same conclusion in the case that|y2| > R, but only on the part of the curve satisfying the hypothesis. We define thenon-empty compact set

B = (0, y2, y3) ∈ Pε : |y2| ≤ R and hε0 ≤ y3 ≤ ρ.

Thus, if p ∈ Dε ∩ (Pε \B), then Λε1(p) = −∞.

4.2 The Aleksandrov function and tilted planes 57

Let us define the sets Dε0 = p ∈ Dε : hε(p) = hε

0, Eε0 = p ∈ E : hε(p) ≥ hε

0and W ε

0 = p ∈ W : hε(p) ≥ hε0, see Figure 4.1.

b

b

∂Eε

0

Eε

0

Dε

Dε

0Πν

Pε

Πνε

nε

p P1(p)

Λε1(p) = −∞

Figure 4.1: The Aleksandrov function Λ1 is valued as −∞ outside of a compact set.

Lemma 4.3. For the end E (with all the assumptions made in this section) and any

plane P parallel to ν = (0, 0, 1), either the function Λ is strictly decreasing, or else M

has a plane of reflection parallel to P.

Proof. We will begin proving that the function Λ is non-increasing. For this it suffi-ces to show that Λ(ρ) ≤ Λ(0) for all ρ > 0, since we can translate vertically E andredefine the end to choose the level ρ = 0 arbitrarily.

Recall that W = C(N) ∩ p ∈ R3 : h(p) ≥ 0. We claim:

Claim A. Λ(ρ) ≤ Λ(0) for all ρ > 0 if, and only if,

E∗t+ ∩ p ∈ R

3 : h(p) > ρ ⊂ W for all t > Λ(0).

Proof. If Λ(ρ) ≤ Λ(0) for all ρ > 0, then for an arbitrary t > Λ(0) wehave that Λ(ρ) ≤ t for all ρ > 0. By the definition of the Aleksandrovfunction, we infer that E∗

t+ ∩ p ∈ R3 : h(p) > ρ ⊂ W .

If E∗t+ ∩ p ∈ R3 : h(p) > ρ ⊂ W for all t > Λ(0), we will show that

Λ(ρ) ≤ Λ(0) for all ρ > 0 reasoning by contradiction. Suppose thereexist ρ0 > 0 and p ∈ D such that h(p) = ρ0 and

Λ1(p) = Λ(ρ0) > Λ(0).

Considering t = Λ(ρ0) we have that if any neighborhood of E∗t+ con-

taining the reflection of the point P1(p) through the plane Pt, wherecontained in W , the maximum principle would yield that Pt is a planeof symmetry for M (as in the proof of Lemma 1.26). But this is impossi-ble since Λ(0) 6= t.


To prove that E∗t+ ∩ p ∈ R3 : h(p) > ρ ⊂ W for all t > Λ(0) we will use tilted

planes and Lemma 1.27.

The semicontinuity of the Aleksandrov function, Lemma 1.27, and Lemma 4.2imply that the maximum value of the function Λε

1 must be achieved at some pointat the compact set Dε ∩ B, where B is the set obtained from Lemma 4.2. We aregoing to prove that the height of this point from the plane Πνε is exactly hε

0.

Claim B. The function Λε1 must attain its maximum at some point in Dε

0.

Proof. Suppose that the function Λε1 attains its maximum value t at some

point q ∈ Dε ∩ B. Note that if p ∈ Dε ∩ (Pε \B) and P1(p) ∈ Et+ then,by Lemma 4.2, the reflection, P1(p)∗, of P1(p) through the plane Pε

t isin W ε

0 . Then we just have to pay attention on the points p ∈ Dε ∩ B

when we reflect through the plain Pεt . Suppose that hε(q) > hε

0, thenthe points P1(q) and P2(q) are far away from ∂Eε

0 , i.e., q is either aninterior tangent point or a boundary tangent point, see Figure 4.2. Inany case, by Lemma 1.26, the set Eε

0 has a plane of symmetry parallelto Pε, which is a contradiction, thereby q ∈ Dε

0.

b

Pε

Pεt

q

W ε0

W ε0

(Eε0)∗

t+

Eε0t+

Figure 4.2: Λε

1 attains its maximum value t at q ∈ Dε ∩B such that hε(q) > hε

0.

Writing zε for the maximum value of Λε1, it follows that

(Eε0)∗

t+ ⊂ W ε0 for all t ≥ zε. (4.4)

By letting ε → 0, from (4.4) we get

E∗t+ ⊂ W for all t ≥ lim sup

ε→0zε. (4.5)

The semicontinuity of the Aleksandrov function, Lemma 1.27, and Claim B impliesthat

lim supε→0

zε ≤ Λ(0).

4.2 The Aleksandrov function and tilted planes 59

Given t ≥ Λ(0), then t ≥ lim supε→0

zε by the inequality above, thus (4.5) says that

E∗t+ ∩ p ∈ R

3 : h(p) > ρ ⊂ W .

Hence, using Claim A, we have proven that the function Λ is non-increasing.

To finish the proof of the lemma, note that the function Λ is either strictly de-creasing or constant in some interval. In the second case, the Aleksandrov functionΛ1 has a local interior maximum and, Lemma 1.26 determines the existence of aplane of symmetry parallel to P.

4.3 A Schoen type theorem for ESWMT-surfaces

In this section we are going to present the main result of this work. We willestablish a Schoen type theorem for embedded ESWMT-surfaces.

Theorem 4.4 (Main Theorem). Let M be a complete connected ESWMT-surface em-

bedded in R3 with finite total second fundamental form and two ends. Suppose that

the function f associated to M is non-negative and Lipschitz. Then M must be rota-

tionally symmetric. In fact, there exists τ > 0 such that M is the surface of revolution

Mτ determined by Sa Earp and Toubiana in [ST95].

Proof. Let E1 and E2 be the ends of M . By Lemma 4.1 we know that E1 and E2 areparallel ends and M grows infinitely in opposite directions. Let ν be the continuousextension of the Gauss map of one of the ends (observe that the Gauss map goesto −ν at the other end) and P a plane parallel to ν; up to a rigid motion we canassume ν = (0, 0, 1). In order to apply Lemma 4.3 we will consider the Aleksandrovfunctions associated to each endE1 andE2 denoted by Λ1 and Λ2 respectively. Theneither both of these Aleksandrov functions are strictly decreasing or at least one ofthe ends has a plane of symmetry parallel to P. In the second case it is clear thatthe geometric comparison principle, Theorem 1.22, implies that M has a plane ofsymmetry parallel to P. For the first case we will consider the Aleksandrov functionassociated to the entire surface M . This function is clearly defined over R since Mgrows infinitely in opposite directions. This function is now strictly increasing onsome interval (−∞, a] and strictly decreasing on some interval [b,∞), which meansthat the maximum value is attained on the compact set [a, b]; which is a globalmaximum, see Figure 4.3. This maximum value must be attained at an interiorpoint and Lemma 1.26 guaranties that M has a plane of symmetry parallel to P.

Therefore, for any vertical plane, P, M is symmetric respect to some plane para-llel to P. Since the center of mass of any cross section must be contained in anyplane of symmetry, then all vertical planes of symmetry intersect in a line parallelto ν. Hence, M is rotationally symmetric respect to this line.


−∞

a

b

ρ Λ(ρ)

+∞

E1

E2

M

Figure 4.3: The associated Aleksandrov function, Λ, on M .

Now we are going to conclude the last part of the theorem. Let f be the ellipticfunction associated to M and observe that f satisfies trivially (2.2). The ellipticitycondition (2.1) implies that 1−2tf ′(t2

)> 0 for all t > 0. Then, the function t−f(t2)

in non-decreasing, hence the following limit exists, possibly +∞,

κ = limt→∞

(t− f

(t2)).

By Theorem 2.6 there is a unique rotationally symmetric f -surface Mτ for eachτ > 0 satisfying 1

τ < κ. By the uniqueness of the ESWMT-surfaces rotationallysymmetric, Theorem 2.10, there exists τ ∈ ( 1

κ ,∞)

such that M = Mτ .

Remark 4.5. By Theorem 2.6, the surface has a horizontal plane of symmetry, which

means that the growth of the two ends are the same; in other words, the constant

β in Theorem 3.21 is the same for both ends up to sign. However, this fact can be

proved using the maximum principle. Let suppose, without loss of generality, that the

growth of the end E1 is bigger than the growth of the end E2, this is, the coefficients

of log r for each end in Theorem 3.21 satisfy |β1| > |β2|, see Figure 4.4. Consider a

horizontal plane low enough and reflect, through this plane, the part of the surface that

remains below the plane; this reflection intersects the surface just on the plane since

|β1| − |β2| > 0, see Figure 4.5. Now we move up the plane until obtain a boundary

tangent point, see Figure 4.6; this point is attained exactly when the Gauss map is

horizontal. By the geometric comparison principle, Theorem 1.22, the horizontal plane

is a plane of symmetry, therefore |β1| = |β2|.

4.3 A Schoen type theorem for ESWMT-surfaces 61

Figure 4.4: A rotational ESWMT-surface with different growth ends.

Figure 4.5: Aleksandrov reflection using a horizontal plane.

Figure 4.6: Finding a boundary tangent point.

As corollary of the main theorem, we can recover the original Schoen theorem,namely

Corollary 4.6. Let M be a complete connected embedded minimal surface in R3 with

finite total curvature and two ends. Then M is a catenoid.

Proof. Since M is minimal, it has finite total curvature if, and only if, it has finitetotal second fundamental form. Then, we can apply Theorem 4.4 to the particularcase f ≡ 0. The conclusion is obtained from the uniqueness of the catenoid as theonly non-planar minimal surface rotationally symmetric.


Bibliography

[Ach92] Naum Achieser. Theory of approximation. Dover publications, inc., 1992 (cit. onp. 37).

[AE07] Juan Aledo and José Espinar. “A conformal representation for linear Weingartensurfaces in the de Sitter space”. In: Journal Geometry and Physics 57 (2007),pp. 1669–1677 (cit. on p. xv).

[AE75] Kinetsu Abe and Joseph Erbacher. “Isometric immersions with the same Gaussmap”. In: Mathematische Annalen 215 (1975), pp. 197–201 (cit. on p. 6).

[Ahl66] Lars Ahlfors. Lectures on Quasiconformal Mappings. D. Van Nostrand Company,Inc, 1966 (cit. on pp. xx, 50).

[AL13] Antonio Alarcón and Francisco López. “On harmonic quasiconformal immer-sions of surfaces in R3”. In: Transactions of the American Mathematical Society

365.4 (2013), pp. 1711–1742 (cit. on pp. 12, 36).

[Ale+07] Hilario Alencar, Manfredo do Carmo, and Renato Tribuzy. “A theorem of Hopfand the Cauchy-Riemann inequality”. In: Communications in Analysis and Geom-

etry 15.2 (2007), pp. 283–298.

[Ale+10] Juan Aledo, José Espinar, and José Gálvez. “The Codazzi equation for surfaces”.In: Advances in Mathematics 224 (2010), pp. 2511–2530 (cit. on pp. xvi, xx, 1,6, 15, 21, 26).

[Ale56] Aleksandr Aleksandrov. “Uniqueness Theorems for Surfaces in the Large, I”. In:Vestnik Leningrad University: Mathematics 11 (1956), pp. 5–17 (cit. on pp. xx,13, 15, 18).

[Ale57] Aleksandr Aleksandrov. “Uniqueness Theorems for Surfaces in the Large, II”. In:Vestnik Leningrad University: Mathematics 12 (1957), pp. 15–44.

[Ale58] Aleksandr Aleksandrov. “Uniqueness Theorems for Surfaces in the Large, V”. In:Vestnik Leningrad University: Mathematics 13 (1958), pp. 5–8.

[AR04] Ube Abresch and Harold Rosenberg. “A Hopf differential for constant mean cur-vature surfaces in S2 × R and H2 × R”. In: Acta Mathematica 193.2 (2004),pp. 141–174 (cit. on p. 6).

[Axl+01] Sheldon Axler, Paul Bourdon, and Ramey Wade. Harmonic Function Theory.Springer-Verlag New York, 2001 (cit. on p. 52).

[Ban80] Victor Bangert. “Total curvature and the topology of complete surfaces”. In: Com-

positio Mathematica 41.1 (1980), pp. 95–105.

63

[Bar+12] Abdênago Barros, Juscelino Silva, and Paulo Sousa. “Rotational Linear Wein-garten Surfaces into the Euclidean Sphere”. In: Israel Journal of Mathematics

192 (2012), pp. 819–830 (cit. on p. xv).

[Ber15] Serge Bernstein. “Sur une théorème de géometrie et ses applications aux équa-tions dérivées partielles du type elliptique”. In: Communications de la Société

Mathématique de Kharkow 15.1 (1915), pp. 38–45 (cit. on pp. xix, 38).

[Ber27] Serge Bernstein. “Über ein geometrisches Theorem und seine Anwendung aufdie partiellen Differentialgleichungen vom elliptischen typus”. In: Mathematis-

che Zeitschrift 26 (1927), pp. 551–558.

[Bry11] Robert Bryant. “Complex analysis and a class of Weingarten surfaces”. arXiv:1105.5589v1 [math.DG]. Preprint. 2011 (cit. on pp. xv, 26).

[BS97] Fabiano Braga and Ricardo Sa Earp. “On the structure of certain Weingartensurfaces with boundary a circle”. In: Annales de la faculté des sciences de Toulouse

6.2 (1997), pp. 243–255 (cit. on pp. xvi, 21, 22).

[Cha00] Hsungrow Chan. “Nonexistence of nonpositively curved surfaces with one em-bedded end”. In: Manuscripta Mathematica 102 (2000), pp. 177–186 (cit. onpp. xix, 36).

[Cha06] Hsungrow Chan. “Simply connected nonpositively curved surfaces in R3”. In:Pacific Journal of Mathematics 223.1 (2006), pp. 1–4.

[Cha10] Hsungrow Chan. “Topological uniqueness of negatively curved surfaces”. In:Nagoya Mathematical Journal 199 (2010), pp. 137–149.

[Che45] Shiing-Shen Chern. “Some new characterizations of the Euclidean sphere”. In:Duke Mathematical Journal 12 (1945), pp. 279–290 (cit. on p. xv).

[Che55a] Shiing-Shen Chern. “An elementary proof of the existence of isothermal parame-ters on a surface”. In: Proceedings of the American Mathematical Society 6 (1955),pp. 771–782 (cit. on p. 5).

[Che55b] Shiing-Shen Chern. “On Special W-Surfaces”. In: Proceedings of the American

Mathematical Society 6.5 (1955), pp. 783–786 (cit. on p. xv).

[Che94] Leung-Fu Cheung. “The Nonexistence of Some Noncompact Constant Mean Cur-vature Surfaces”. In: Proceedings of the American Mathematical Society 121.4(1994), pp. 1207–1209 (cit. on p. 12).

[CM02] Tobias Colding and William Minicozzi II. “Estimates for Parametric Elliptic Inte-grands”. In: International Mathematics Research Notices 6 (2002), pp. 291–297.

[CM11] Tobias Colding and William Minicozzi II. A course in minimal surfaces. AmericanMathematical Society, 2011 (cit. on p. 12).

[CM99] Tobias Colding and William Minicozzi II. Minimal surfaces. Vol. 4. New YorkUniversity Courant Institute of Mathematical Sciences, 1999 (cit. on p. 12).

[CO67] Shiing-Shen Chern and Robert Osserman. “Complete minimal surfaces in eu-clidean n-space”. In: Journal d’Analyse Mathématique 19.1 (1967), pp. 15–34(cit. on pp. xvii, xviii).

64 Bibliography

[Con+15] Peter Connor, Kevin Li, and Matthias Weber. “Complete Embedded HarmonicSurfaces in R3”. In: Experimental Mathematics 24.2 (2015), pp. 196–224 (cit.on pp. xix, xxi, 36).

[CT01] Hsungrow Chan and Andrejs Treibergs. “Nonpositively curved surfaces in R3”.In: Journal of Differential Geometry 57 (2001), pp. 389–407 (cit. on pp. 36, 37).

[CV33] Stephan Cohn-Vossen. “Sur la courbure totales des surfaces ouvertes”. In: Comptes

Rendus de l’Académie des Sciences 197 (1933), pp. 1165–1167 (cit. on p. 11).

[Daj90] Marcos Dajczer. Submanifolds and isometric immersions. Publish or Perish, Incor-porated, 1990 (cit. on p. 3).

[de +09] Sebastião de Almeida, Fabiano Brito, and Roseli Nozaki. “Closed Special Wein-garten Surfaces in the Standard Three Sphere”. In: Results in Mathematics 56(2009), pp. 501–518 (cit. on p. xvi).

[do 76] Manfredo do Carmo. Differential geometry of curves and surfaces. Prentice-Hall,1976 (cit. on pp. 3, 6).

[do 92] Manfredo do Carmo. Riemannian geometry. Birkhäuser, 1992.

[Esp08] José Espinar. “La ecuación de Codazi en superficies”. PhD thesis. Universidadde Granada, 2008 (cit. on pp. 1, 6, 15).

[Eva98] Lawrence Evans. Partial differential equations. Vol. 19. Graduate studies in math-ematics. American Mathematical Society, 1998 (cit. on p. 48).

[Fin64] Robert Finn. “On normal metrics, and a theorem of Cohn-Vossen”. In: Bulletin

of the American Mathematical Society 70 (1964), pp. 772–773 (cit. on p. 49).

[Fin65] Robert Finn. “On a class of conformal metrics, with applications to differen-tial geometry in the large”. In: Commentarii Mathematici Helvetici 40.1 (1965),pp. 1–30 (cit. on pp. 11, 49).

[FP17] Abigail Folha and Carlos Peñafiel. “Weingarten Type Surfaces in H2 × R andS2 × R”. In: Canadian Journal of Mathematics 69.6 (2017), pp. 1292–1311.

[GM00] José Gálvez and Antonio Martínez. “The Gauss map and second fundamentalform of surfaces in R3”. In: Geometriae Dedicata 81 (2000), pp. 181–192 (cit.on pp. xix, 6, 29).

[GM18] Antonio Gálvez and Pablo Mira. “Rotational symmetry of Weingarten spheres inhomogeneous three-manifolds”. arXiv: 1807.09654 [math.DG]. Preprint. 2018.

[Grö28] Herbert Grötzsch. “Über einige Extremalprobleme der konformen Abbildung. I,II.” In: Berichte Leipzig 80 (1928), pp. 376–376, 497–502 (cit. on p. xx).

[GT01] David Gilbarg and Neil Trudinger. Elliptic partial differential equations of second

order. Springer-Verlag Berlin Heidelberg, 2001 (cit. on p. 48).

[Gut+12] Vladimir Gutlyanskii, Vladimir Ryazanov, Uri Srebro, and Eduard Yakubov. The

Beltrami Equation. A Geometric Approach. Springer-Verlag New York, 2012 (cit.on p. 50).

[Gá+03] José Gálvez, Antonio Martínez, and Francisco Milan. “Linear Weingarten sur-faces in R3”. In: Monatshefte für Mathematik 138 (2003), pp. 133–144 (cit. onp. xv).

Bibliography 65

[Gá+15] José Gálvez, Antonio Martínez, and José Teruel. “Complete surfaces with endsof non positive curvature”. In: Advances in Mathematics 281 (2015), pp. 1202–1215.

[Gá+16] José Gálvez, Antonio Martínez, and José Teruel. “Complete surfaces with endsof non positive curvature”. arXiv: 1405.0851 [math.DG]. Preprint. 2016 (cit. onp. xvi).

[Hau+15] Laurent Hauswirth, Barbara Nelli, Ricardo Sa Earp, and Eric Toubiana. “A Schoentheorem for minimal surfaces in H2 × R”. In: Advances in Mathematics 274(2015), pp. 199–240 (cit. on p. xviii).

[HM90] David Hoffman and William Meeks III. “The strong halfspace theorem for min-imal surfaces”. In: Inventiones Mathematicae 101 (1990), pp. 373–377 (cit. onp. 24).

[HO83] David Hoffman and Robert Osserman. “The Gauss map of surfaces in Rn”. In:Journal of Differential Geometry 18 (1983), pp. 733–754 (cit. on p. 6).

[HO85] David Hoffman and Robert Osserman. “The Gauss map of surfaces in R3 andR4”. In: Proceedings of the London Mathematical Society 50.3 (1985), pp. 27–56(cit. on p. 6).

[Hop27] Eberhard Hopf. “Elementäre Bemerkungen über die Lösungen partieller Differ-entialgleichungen zweiter Ordnung vom elliptischen Typus”. In: Sitzungsberichte

Preussische Akademie der Wissenschaften 19 (1927), pp. 147–152 (cit. on p. 13).

[Hop50a] Eberhard Hopf. “A theorem on the accessibility of boundary parts of an openpoint set”. In: Proceedings of the American Mathematical Society 1 (1950), pp. 76–79 (cit. on pp. xix, 38).

[Hop50b] Eberhard Hopf. “On S. Bernstein’s theorem on surfaces z(x, y) of nonpositivecurvature”. In: Proceedings of the American Mathematical Society 1 (1950), pp. 80–85 (cit. on pp. xix, 36).

[Hop52] Eberhard Hopf. “A remark on linear elliptic differential equations of secondorder”. In: Proceedings of the American Mathematical Society 3.5 (1952), pp. 791–793 (cit. on p. 13).

[Hop83] Heinz Hopf. Differential geometry in the large. Vol. 1000. Springer-Verlag BerlinHeidelberg, 1983 (cit. on pp. xv, 6, 18).

[HT92] Dominique Hulin and Marc Troyanov. “Prescribing curvature on open surfaces”.In: Mathematische Annalen 293 (1992), pp. 277–315 (cit. on pp. xviii, 11).

[Hub57] Alfred Huber. “On subharmonic functions and differential geometry in the large”.In: Commentarii Mathematici Helvetici 32 (1957), pp. 13–72 (cit. on pp. xviii,11).

[Hub66] Alfred Huber. “Vollständige konforme metriken und isolierte singularitäten sub-harmonischer funktionen”. In: Commentarii Mathematici Helvetici 41 (1966),pp. 105–136 (cit. on p. 11).

[HW54] Philip Hartman and Aurel Wintner. “Umbilical Points and W-Surfaces”. In: Amer-

ican Journal of Mathematics 76.3 (1954), pp. 502–508 (cit. on p. xv).

66 Bibliography

[JM83] Luquesio Jorge and William Meeks III. “The topology of complete minimal sur-faces of finite total Gaussian curvature”. In: Topology 22.2 (1983), pp. 203–221(cit. on pp. xvii, xviii).

[Ken79] Katsuei Kenmotsu. “Weierstrass formula for surfaces of prescribed mean curva-ture”. In: Mathematische Annalen 245 (1979), pp. 89–99 (cit. on pp. xix, xx,xxiii, 1, 6, 8, 10).

[Kor+89] Nicholas Korevaar, Rob Kusner, and Bruce Solomon. “The structure of completeembedded surfaces with constant mean curvature”. In: Journal of Differential

Geometry 30 (1989), pp. 465–503 (cit. on pp. xvi, xx, 1, 15).

[LT91] Peter Li and Luen-Fai Tam. “Complete surfaces with total finite curvature”. In:Journal of Differentia Geometry 33 (1991), pp. 139–168 (cit. on p. 11).

[Ló08a] Rafael López. “On linear Weingarten surfaces”. In: International Journal of Math-

ematics 19.4 (2008), pp. 439–448.

[Ló08b] Rafael López. “Rotational linear Weingarten surfaces of hyperbolic type”. In:Israel Journal of Mathematics 167 (2008), pp. 283–301 (cit. on p. xv).

[Mil80] Tilla Milnor. “Abstract Weingarten surfaces”. In: Journal of Differential Geometry

15 (1980), pp. 365–380 (cit. on pp. 1, 20, 27).

[Mor56] Akira Mori. “On an absolute constant in the theory of quasi-conformal map-pings”. In: Journal of the Mathematical Society of Japan 8.2 (1956), pp. 156–166 (cit. on pp. xx, 50).

[MR13] Filippo Morabito and Magdalena Rodríguez. “Classification of rotational specialWeingarten surfaces of minimal type in S2 × R and H2 × R”. In: Mathematische

Zeitschrift 273 (2013), pp. 379–399 (cit. on p. xvi).

[MŠ95] Stefan Müller and Vladimír Ševerák. “On surfaces of finite total curvature”. In:Journal of Differentia Geometry 42 (1995), pp. 229–258 (cit. on pp. xix, 12).

[Oss64] Robert Osserman. “Global properties of minimal surfaces in E3 and En”. In:Annals of Mathematics 80.2 (1964), pp. 340–364 (cit. on pp. xviii, 12).

[Oss69] Robert Osserman. A survey of minimal surfaces. Van Nostrand Reinhold Com-pany, 1969 (cit. on pp. xviii, 6, 7, 10, 11).

[Oss71] Robert Osserman. “The convex hulll property of immersed manifolds”. In: Jour-

nal of Differential Geometry 6.2 (1971), pp. 267–270.

[Pal88] Bennett Palmer. “Isothermic surfaces and the Gauss map”. In: Proceedings of the

American Mathematical Society 104.3 (1988), pp. 876–884.

[PS07] Patrizia Pucci and James Serrin. The Maximum Principle. Birkhäuser Basel, 2007(cit. on pp. 1, 13).

[RS94] Harold Rosenberg and Ricardo Sa Earp. “The geometry of properly embeddedspecial surfaces in R3; e.g., surfaces satisfying aH + bK = 1, where a and b

are positive”. In: Duke Mathematical Journal 73.2 (1994), pp. 291–306 (cit. onpp. xv, xvi, 21, 22).

[Sa +91] Ricardo Sa Earp, Fabiano Brito, William Meeks III, and Harold Rosenberg. “Struc-ture Theorems for Constant Mean Curvature Surfaces Bounded by a PlanarCurve”. In: Indiana University Mathematics Journal 40.1 (1991), pp. 333–343.

Bibliography 67

[Sch83] Richard Schoen. “Uniqueness, symmetry, and embeddedness of minimal sur-faces”. In: Journal of Differential Geometry 18 (1983), pp. 791–809 (cit. onpp. xvii, xviii, xx, 33, 34, 53).

[Shi85] Katsuhiro Shiohama. “Total curvatures and minimal areas of complete opensurfaces”. In: Proceedings of the American Mathematical Society 94.2 (1985),pp. 310–316 (cit. on p. 11).

[ST01] Ricardo Sa Earp and Eric Toubiana. “Variants on Alexandrov reflection principleand other applications of maximum principle”. In: Séminaire de Théorie spectrale

et géométrie 19 (2001), pp. 93–121 (cit. on pp. xvi, 21).

[ST93] Ricardo Sa Earp and Eric Toubiana. “A note on special surfaces in R3”. In:Matemática Contemporânea 4 (1993), pp. 108–118 (cit. on pp. xvi, 21, 22).

[ST95] Ricardo Sa Earp and Eric Toubiana. “Sur les surfaces de Weingarten spéciales detype minimal”. In: Boletim da Sociedade Brasileira de Matemática 26.2 (1995),pp. 129–148 (cit. on pp. ix, xi, xvi, xx, xxii, xxiii, 21, 23–25, 55, 60).

[ST99a] Ricardo Sa Earp and Eric Toubiana. “Classification des surfaces de type Delau-nay”. In: American Journal of Mathematics 121.3 (1999), pp. 671–700.

[ST99b] Ricardo Sa Earp and Eric Toubiana. “Symmetry of properly embedded specialWeingarten surfaces in H3”. In: Transactions of the American Mathematical Soci-

ety 351.12 (1999), pp. 4693–4711 (cit. on pp. xvi, 21).

[Tal01a] Steven Taliaferro. “Isolated singularities of nonlinear elliptic inequalities”. In:Indiana University Mathematics Journal 50.4 (2001), pp. 1885–1898 (cit. onp. xx).

[Tal01b] Steven Taliaferro. “On the growth of superharmonic functions near isolated sigu-larity, II”. In: Communications in Partial Differential Equations 26.5-6 (2001),pp. 1003–1026.

[Tal06] Steven Taliaferro. “Isolated singularities of nonlinear elliptic inequalities II. Asymp-totic behavior of solutions”. In: Indiana University Mathematics Journal 55.6(2006), pp. 1791–1812 (cit. on pp. 46, 49).

[Tal99] Steven Taliaferro. “On the growth of superharmonic functions near isolated sigu-larity, I”. In: Journal of Differential Equations 158 (1999), pp. 28–47 (cit. onpp. xx, 46, 48).

[Wei61] Julius Weingarten. “Ueber eine Klasse auf einander abwickelbarer Flächen”. In:Journal für die Reine und Angewandte Mathematik 59 (1861), pp. 382–393 (cit.on p. xv).

[Wei63] Julius Weingarten. “Ueber die Flächen deren Normalen eine gegebene Flächeberühren”. In: Journal für die Reine und Angewandte Mathematik 62 (1863),pp. 61–63 (cit. on p. xv).

[Whi87] Brian White. “Complete surfaces of finite total curvature”. In: Journal of Differ-

ential Geometry 26 (1987), pp. 315–326 (cit. on pp. xix, xxii, 1, 11, 12).

[Whi88] Brian White. “Correction to Complete surfaces of finte total curvature”. In: Jour-

nal of Differential Geometry 28 (1988), pp. 359–360.

68 Bibliography

Declaration

I, Héber Mesa Palomino, confirm that the work presented here was solely under-taken by myself and that no help was provided from other sources as those allowed.All sections of the paper that use quotes or describe an argument or concept de-veloped by another author have been referenced, including all secondary literatureused, to show that this material has been adopted to support my thesis.

Rio de Janeiro, RJ-Brazil, April, 2019


elliptic special weingarten surfaces of minimal type with ... · este trabajo no hubiese sido...

Documents